MATH2210 Notebook 1 Fall Semester 2016/ MATH2210 Notebook Solving Systems of Linear Equations... 3


 Edgar Boone
 2 years ago
 Views:
Transcription
1 MATH0 Notebook Fall Semester 06/07 prepared by Professor Jenny Baglivo c Copyright by Jenny A. Baglivo. All Rights Reserved. Contents MATH0 Notebook 3. Solving Systems of Linear Equations Linear Equations and Linear Systems Solutions and Solution Sets Elementary Row Operations, Forward and Backward Pass Existence and Uniqueness Geometric Interpretations Row Reductions and Echelon Forms Echelon Form and Reduced Echelon Form Pivot Positions, Pivot Columns and Pivots Basic and Free Variables; Solutions to Linear Systems Vectors and Vector Equations Vectors in mdimensional Space Vector Sum, Scalar Product, and Linear Combinations of Vectors Linear Combinations and Spans Vector Equations, Spans and Solution Sets Matrix Equations and Solution Sets Coefficient Matrix and MatrixVector Product Matrix Equation, Span and Consistency Solution Sets and Homogeneous Systems Solution Sets and Nonhomogeneous Systems Linearly Independent and Linearly Dependent Sets Linear and Matrix Transformations
2 .6. Transformations, Images and PreImages OneToOne and Onto Transformations Linear Transformations Linear and Matrix Transformations
3 MATH0 Notebook This notebook is concerned with introductory linear algebra concepts. The notes correspond to material in Chapter of the Lay textbook.. Solving Systems of Linear Equations In linear algebra we study linear equations and systems of linear equations. Linear algebra methods are used to solve problems in areas as diverse as ecology (e.g. population projections), economics (e.g. inputoutput analysis), engineering (e.g. analysis of air flow), and computer graphics (e.g. perspective drawing). Further, linear methods are fundamental in statistical analysis of multivariate data... Linear Equations and Linear Systems A linear equation in the variables x, x,..., x n is an equation of the form a x + a x + + a n x n = b where a, a,..., a n and b are constants. A system of linear equations in the variables x, x,..., x n is a collection of one or more linear equations in these variables. For example, x x + x 3 = 0 x 8x 3 = 8 4x +5x +9x 3 = 9 is a system of 3 linear equations in the 3 unknowns x, x, x 3 (a 3by3 system )... Solutions and Solution Sets A solution of a linear system is a list (s, s,..., s n ) of numbers that makes each equation in the system true when s i is substituted for x i for i =,,..., n. The list (9, 6, 3) is a solution to the system above. To check this, note that (9) (6) + (3) = 0 (6) 8(3) = 8 4(9) +5(6) +9(3) = 9 The solution set of a linear system is the collection of all possible solutions of the system. For the example above, (9, 6, 3) is the unique solution. Two linear systems are equivalent if each has the same solution set. 3
4 ..3 Elementary Row Operations, Forward and Backward Pass The strategy for solving a linear system is to replace the system with an equivalent one that is easier to solve. The equivalent system is obtained using elementary row operations:. (Replacement) Add to one row a multiple of another,. (Interchange) Interchange two rows, 3. (Scaling) Multiply all entries in a row by a nonzero constant, where each row corresponds to an equation in the system. Example. We work out the strategy using the 3by3 system given above, where a 3by4 augmented matrix of the coefficients and right hand side values follows our progress toward a solution. R : x x + x 3 = 0 0 () R : x 8x 3 = R 3 : 4x +5x +9x 3 = () (3) (4) (5) (6) R : R : R 3 + 4R : R : R : R 3 : R : R : R 3 + 3R : R R 3 : R + 4R 3 : R 3 : R + R : R : R 3 : x x + x 3 = 0 x 8x 3 = 8 3x +3x 3 = 9 x x + x 3 = 0 x 4x 3 = 4 3x +3x 3 = 9 x x + x 3 = 0 x 4x 3 = 4 x 3 = 3 x x = 3 x = 6 x 3 = 3 x = 9 x = 6 x 3 = 3 Thus, the solution is x = 9, x = 6, and x 3 = Forward and backward phases. In the forward phase of the row reduction process (systems () through (4)), R is used to eliminate x from R and R 3 ; then R is used to eliminate x from R 3. In the backward phase of the row reduction process (systems (5) and (6)), R 3 is used to eliminate x 3 from R and R ; then R is used to eliminate x from R. 4
5 ..4 Existence and Uniqueness Two fundamental questions about linear systems are:. Does a solution exist?. If a solution exists, is it unique? If a linear system has one or more solutions, then it is said to be consistent; otherwise, it is said to be inconsistent. A linear system can be inconsistent, or have a unique solution, or have infinitely many solutions. Note that in the example above, we knew that the system was consistent once we reached equivalent system (4); the remaining steps allowed us to find the unique solution. Example. The following 3by3 linear system has an infinite number of solutions: x +4x +x 3 = 6 3x +7x +x 3 = 9 4x x = 5 To demonstrate this, consider the sequence of equivalent augmented matrices: The last matrix corresponds to x + 3x 3 =, x x 3 = 5, and 0 = 0. Thus, ( 3x 3, 5 + x 3, x 3 ) is a solution for any value of x 3. Footnote on Example : The elementary row operations used in this example were R R R 3 5
6 Example 3. The following 3by3 linear system is inconsistent: 3x 6x 3 = 8 x x +3x 3 = 5x 7x +9x 3 = 0 To demonstrate this, consider the sequence of equivalent augmented matrices: The last matrix corresponds to x x + 3x 3 =, 3x 6x 3 = 8, and 0 = 3. Since 0 3, the system has no solution. Footnote on Example 3: The elementary row operations used in this example were R R R 3 Example 4. The following 3by4 system has an infinite number of solutions: x + x x 3 = 9 x +x 3x 3 4x 4 = 9 3x x x +x 3 3x 4 = 3 To demonstrate this, consider the sequence of equivalent augmented matrices: The last matrix corresponds to x + 3x 4 = 5, x x 4 = 8, and x 3 + x 4 = 4. Thus, (5 3x 4, 8 + x 4, 4 x 4, x 4 ) is a solution for any value of x 4. Footnote on Example 4: The elementary row operations used in this example were R R R 3 6
7 Example 5. The following 3by system is inconsistent: x +4x = 6 3x 5x = 44 3x x = 55 To demonstrate this, consider the sequence of equivalent augmented matrices: The last matrix corresponds to x + x = 3, x = 5, and 0 = 4. Since 0 4, the system has no solution. Footnote on Example 5: The elementary row operations used in this example were R R R 3..5 Geometric Interpretations Linear equations in two variables correspond to lines in the plane. Linear equations in three variables correspond to planes in 3space. Thus, solution sets for mby and mby3 systems have natural geometric interpretations. Example 6. Consider the following three by systems: () x + x = 5 x x = 0 () x +x = 3 x 4x = 8 (3) x + x = 3 x x = 6. System () has the unique solution (.5,.5), the point of intersection of the lines.. System () is inconsistent; the lines are unequal and parallel. 3. System (3) has an infinite number of solutions: (3 x, x ) for any value of x, since the two equations graph to the same line. 7
8 Example, revisited. Each pair of planes intersect in a line. The lines are quite close in 3space. In addition, the three planes have a single point of intersection, (9, 6, 3).. Row Reductions and Echelon Forms.. Echelon Form and Reduced Echelon Form A matrix is in (row) echelon form when. All nonzero rows are above any row of zeros.. Each leading entry (that is, leftmost nonzero entry) in a row is in a column to the right of the leading entries of the rows above it. 3. All entries in a column below a leading entry are zero. An echelon matrix is in reduced form when (in addition) 4. The leading entry in each nonzero row is. 5. Each leading is the only nonzero entry in its column. Example. The 6by matrix shown on the left below is in echelon form and the 6by matrix shown on the right is in reduced echelon form In the left matrix, the symbol represents a leading nonzero entry and the symbol represents any number (either zero or nonzero). In the right matrix, each leading entry has been converted to a, and each is the only nonzero entry in its column. 8
9 Starting with the augmented matrix of a system of linear equations, the forward phase of the row reduction process will produce a matrix in echelon form. Continuing the process through the backward phase, we get a reduced echelon form. The following theorem explains why reduced echelon form matrices are important. Theorem (Uniqueness Theorem). Each matrix is rowequivalent (that is, equivalent using elementary row operations) to one and only one reduced echelon form matrix. Further, we can read the solutions to the original system from the reduced echelon form of the augmented matrix... Pivot Positions, Pivot Columns and Pivots. A pivot position is a position of a leading entry in an echelon form matrix.. A column that contains a pivot position is called a pivot column. 3. A pivot is a nonzero number that either is used in a pivot position to create zeros or is changed into a leading, which in turn is used to create zeros. In general, there is no more than one pivot in any row and no more than one pivot in any column. Example, continued. In the left matrix in Example, each is located at a pivot position, and columns, 4, 5, 8, and 9 are pivot columns. Example. Consider the following 4by5 matrix We first interchange R and R 4 : Column is a pivot column, and the in the upper left corner is used to change values in rows and 3 to 0 (R + R replaces R ; R 3 + R replaces R 3 ):
10 Column is a pivot column. We could interchange any of the remaining three rows to identify a pivot, but will just use the in the current second row to change values in rows 3 and 4 to 0 (R 3 5 R replaces R 3 ; R R replaces R 4 ): Column 3 is not a pivot column. Column 4 is a pivot column, revealed after we we interchange R 3 and R 4 : The matrix is now in echelon form. Columns,, and 4 are pivot columns. Working with rows 3 and (in that order) to clear columns 4 and, we get the following reduced echelon form matrix: Note that if the original matrix was the augmented matrix of a 4by4 system of linear equations in x, x, x 3, x 4, then the last matrix corresponds to the equivalent linear system x 3x 3 = 5, x + x 3 = 3, x 4 = 0, 0 = 0. Thus, (5 + 3x 3, 3 x 3, x 3, 0) is a solution for any value of x 3. Footnote on Example : The last elementary row operations were: R R R 3 R 4 0
11 Example 3. Consider the following 3by6 matrix: We first interchange R and R 3 : Column is a pivot column and we use the 3 in the upper left corner to convert the 3 in row to 0 (R R replaces R ). In addition, R is replaced by 3 R for simplicity: Column is a pivot column. We could use the current row, or interchange the second and third rows and use the new row two for pivot. For simplicity, we use the in the second row to convert the 3 in the third row to a zero (R 3 3 R replaces R 3 ). In addition, R is replaced by R for simplicity: This matrix is in echelon form; columns,, and 5 are pivot columns. Working with rows 3 and (in that order) to clear columns 5 and, we get the following reduced echelon form matrix: Note that if the original matrix was the augmented matrix of a 3by5 system of linear equations in x, x, x 3, x 4, x 5, then the last matrix corresponds to the equivalent linear system x x 3 + 3x 4 = 4, x x 3 + x 4 = 7, x 5 = 4. Thus, ( 4 + x 3 3x 4, 7 + x 3 x 4, x 3, x 4, 4) is a solution for any values of x 3, x 4. Footnote on Example 3: The last elementary row operations were: R R R 3
12 ..3 Basic and Free Variables; Solutions to Linear Systems When solving linear systems,. A basic variable is any variable that corresponds to a pivot column in the augmented matrix of the system, and a free variable is any nonbasic variable.. From the equivalent system produced using the reduced echelon form, we solve each equation for the basic variable in terms of the free variables (if any) in the equation. 3. If there is at least one free variable, then the original linear system has an infinite number of solutions. 4. If an echelon form of the augmented matrix has a row of the form [ b ] where b 0, then the system is inconsistent. (There is no need to continue to find the reduced echelon form.) Example, continued. For the 4by4 system in Example, the basic variables are, and the free variables are. Example 3, continued. For the 4by5 system in Example 3, the basic variables are, and the free variables are..3 Vectors and Vector Equations.3. Vectors in mdimensional Space In linear algebra, we let R m be the set of m matrices of real numbers, Further, v R m v =. v m v : v v i R, and we let v =. Rm be a specific element. v m. v is known as a column vector or simply a vector.. The value v i is the i th component of v. 3. The zero vector, O, is the vector all of whose components equal zero.
13 .3. Vector Sum, Scalar Product, and Linear Combinations of Vectors Suppose v, w R m and c, d R.. The vector sum v + w is the vector obtained by componentwise addition. That is, v + w is the vector whose i th component is v i + w i for each i.. The scalar product cv is the vector obtained by multiplying each component by c. That is, cv is the vector whose i th component is cv i for each i. 3. The vector cv + dw is known as a linear combination of the vectors v and w. [ ] [ ] Example in space. Let v = and v = be vectors in R. These vectors can be represented as points in the plane or as directed line segments whose initial point is the origin. In the plot, Vector v is represented as the directed line segment from (0, 0) to (, ), and points on y = x represent scalar multiples of v : cv, where c R. Vector v is represented as the directed line segment from (0, 0) to (, ), and points on y = x represent scalar multiples of v : dv, where d R Linear combinations of the two vectors, cv +dv, are in onetoone correspondence with R. The plot shows a grid of linear combinations where either c or d is an integer. Problem. Write w = [ ] 7 as a linear combination of v 5 and v. 3
14 Example in 3space. Let v = and v = The thick lines represent scalar multiples of the two vectors: cv for c R, and dv for d R. The plane shown represents the linear combinations of the two vectors: cv + dv, for c, d R. The grid represents the linear combinations where either c or d is an integer. Problem. Using v and v from above: be vectors in R 3. x (a) Suppose that y = cv + dv for some c, d R. Find an equation relating x, y, z. z (b) Can w = 3 be written as a linear combination of v and v? Why? 0 4
15 .3.3 Linear Combinations and Spans If v, v,..., v n R m and c, c,..., c n R, then w = c v + c v + + c n v n is a linear combination of the vectors v, v,..., v n. The span of the set {v, v,..., v n } is the collection of all linear combinations of the vectors v, v,..., v n : Span{v, v,..., v n } = {c v + c v + + c n v n : c i R for each i} R m. The span of a set of vectors always contains the zero vector O (let each c i = 0). For example, {[. Span ]} = { [ c ] } : c R corresponds to the line y = x in R. {[. Span ] [, ]} = { [ c ] [ + d R can be written as a linear combination of ] } : c, d R = R. That is, every vector in [ ] [ ] and. {[ 3. Span ] [, ] [ 7, 5 ]} = { [ c ] [ + d ] [ 7 + e 5 ] } : c, d, e R = R. That is, every vector in R can be written as a linear combination of the three listed vectors. (In fact, the first two vectors were already sufficient to represent each v R.) 4. Span, = c + d : c, d R corresponds to the plane with equation in R Span,, 4 0 = Span in the list is a linear combination of the first two., since the third vector 5
16 Exercise. Let v =, v = and v 3 = 3. Demonstrate that 0 Span{v, v, v 3 } = {cv + dv + ev 3 : c, d, e R} = R 3. That is, demonstrate that every vector in R 3 can be written as a linear combination of the three vectors v, v, v 3. 6
17 .3.4 Vector Equations, Spans and Solution Sets Consider an mbyn system of equations in variables x, x,..., x n, where each equation is of the form a i x + a i x + + a in x n = b i for i =,,..., m. Let a j be the vector of coefficients of x j, and b be the vector of right hand side values. The mbyn system can be rewritten as a vector equation: x a + x a + + x n a n = b. The system is consistent if and only if b Span{a, a,..., a n }. For example, the linear system on the left below becomes the vector equation on the right: x +5x 3x 3 = 4 x 4x + x 3 = 3 x 7x = = x + x + x 3 =. Problem. Starting with the vector equation x a + x a + x 3 a 3 = b shown on the right above, (a) Determine if the system is consistent. That is, determine if b Span {a, a, a 3 }. x (b) Suppose that y z = ca + da + ea 3 for some c, d, e. Find an equation relating x, y, z. 7
18 (c) Find the reduced echelon form of the matrix whose columns are the a i s: A = [ a a a 3 ]. (d) What do we know about the set Span{a, a, a 3 }? Be as complete as possible. 8
19 .4 Matrix Equations and Solution Sets As above, an mbyn system of linear equations, a i x + a i x + + a in x n = b i for i =,,..., m, leads to a vector equation, x a + x a + + x n a n = b, where a j is the vector of coefficients of x j, and b is the vector of right hand side values. Further, the system is consistent if and only if b Span{a,..., a n }..4. Coefficient Matrix and MatrixVector Product. The coefficient matrix A of an mbyn system of linear equations is the m n matrix whose columns are the a j s: A = [ a a a n ].. If A is an m n matrix and v R n, then the matrixvector product of A with v is defined to be the the following linear combination of the columns of A: v Av = [ ] v a a a n. = v a + v a + + v n a n. v n For example, if A = Av = 3 and v = , then 3 0 =. Rowcolumn definition. The definition of the matrixvector product as a linear combination of column vectors is equivalent to the usual rowcolumn definition of the product of two matrices. For example, (3) + 5( ) 3() Av = 4 = (3) 4( ) + () =. 7 0 (3) 7( ) + 0() Linearity property. The matrixvector product of a linear combination of k vectors in R n is equal to that linear combination of matrixvector products: A (c v + c v + + c k v k ) = c Av + c Av + + c k Av k for all c i R, v i R n. That is, multiplication by A distributes over addition and we can factor out constants. 9
20 .4. Matrix Equation, Span and Consistency Let A = [ a a a n ] be the coefficient matrix of an mbyn system of linear equations, x R n be the vector of unknowns, and b R m be the vector of right hand side values. Then. The matrix equation of the system is the equation Ax = b.. If Ax = b is consistent for all b, then the matrix A is said to span R m. For example, the linear equation on the left becomes the matrix equation Ax = b on the right: x +5x 3x 3 = x 4 x 4x + x 3 = 3 = 4 x = 3. x 7x = 7 0 x 3 Further, since the system is inconsistent, A does not span R 3. The following theorem gives criteria for determining when A spans R m. Theorem (Consistency Theorem). Let A = [ a a a n ] be an m n coefficient matrix, and x R n be a vector of unknowns. The following statements are equivalent: (a) The equation Ax = b has a solution for all b R m. (b) Each b R m is a linear combination of the coefficient vectors a j. (c) The columns of the matrix A span R m. That is, Span{a, a,..., a n } = R m. (d) A has a pivot position in each row. Problem. Let A = () Determine if Ax = b is consistent for all b R 3. () If it is not consistent for all b, find an equation characterizing when Ax = b is consistent. 0
21 Problem. Let A = 4 0. () Determine if Ax = b is consistent for all b R 3. () If it is not consistent for all b, find an equation characterizing when Ax = b is consistent. Problem 3. A matrix can have at most pivot position in any row or any column. What, if anything, does this fact tell you about determining consistency for all b when m < n, m = n, m > n? Be as complete as possible in your explanation.
22 .4.3 Solution Sets and Homogeneous Systems Let Ax = b be the matrix equation of an mbyn system of linear equations.. The solution set of the system is {x : Ax = b} R n. A solution set can be empty, contain one element or contain an infinite number of elements.. If b = O (the zero vector), then the system is said to be homogeneous. Homogeneous systems satisfy the following properties:. Every homogeneous system is consistent.. If v, v,..., v k are solutions to the homogeneous system and c, c,..., c k are constants, then the linear combination is also a solution to the system. w = c v + c v + + c k v k 3. The solution set of Ax = O can be written as the span of a certain set of vectors. Problem 4. Demonstrate the first two properties of homogeneous systems.
23 Writing the solution set of Ax=O as a span. The technique for writing the solution set of Ax = O is illustrated using the following example: Consider the homogeneous system x +5x 3x 3 = x 0 x 4x + x 3 = 0 = 4 x = 0. x 7x = x 3 0 Since x = 7x 3 x = x 3 x 3 free, we can write the solutions to the homogeneous system as vectors satisfying x = 7x x 3 = x 3 where x 3 is free. Thus, Span = c x 3 is the solution set of the homogeneous system above. 7 : c R Problem 5. In each case, write the solution set of Ax = O as a span of a set of vectors. (a) Let A =. 8 (b) Let A = [ 8 4 ]. 3
24 (c) Let A = Writing the solution set of Ax=O in parametric vector form. If the solution set to Ax = O is Span{v, v,..., v k }, then the solutions can be written in parametric vector form as follows: x = c v + c v c k v k, where c, c,..., c k R, and the c i s are the parameters in the general equation for the solution. For example, the parametric vector form for the solutions in Problem 5(b) is x = c, for c R (only one parameter is needed); and the parametric vector form for the solutions in Problem 5(c) is x = c + c, for c, c R (two parameters are needed). 4
25 .4.4 Solution Sets and Nonhomogeneous Systems A linear system is nonhomogeneous if it is not homogeneous. That is, a linear system is nonhomogeneous if its matrix equation is of the form Ax = b, where b O. Theorem (Solution Sets). Suppose the equation Ax = b is consistent and let p be a particular solution. Then the solution set of Ax = b is the set of all vectors of the form x = p + v h, where v h is a solution to Ax = O. That is, each solution can be written as the vector sum of a particular solution to the nonhomogeneous system and a solution to the homogeneous system with the same coefficient matrix. For example, let [ [ A b ] = [ x = ] and b = ] 6 x x we can write x = p + v h, where p = [ 0 4 ]. Since [ = ] + x [ and v h = x We can visualize the solution sets to both systems as follows: Solid Line: The solution to the homogeneous system is the solid line c 0, for some c. Dashed Line: The solution to the original nonhomogeneous system is the dashed line c, for some c. 0 ], where x is free, 0 for some x. (The solutions are written in parametric vector form, using c as the parameter.) x = 6 x x is free x 3 = Each point on the dashed line is obtained by adding p to a point on the solid line. The lines are parallel and lie on the plane x + y 3z = 0. Shift of a span. Since the set of solutions to Ax = O is the span of a set of vectors, the set of solutions to Ax = b can be described as the shift of a span, where each solution of the homogeneous system is shifted by the particular solution p. 5
26 Problem 6. Let A = and b = 8 4 Write the solutions to Ax = b in parametric vector form. (That is, write the solutions in the form x = p + v h, where p is a particular solution to the nonhomogeneous system and v h is a general solution to the homogeneous system with the same coefficient matrix.). 6
27 .5 Linearly Independent and Linearly Dependent Sets The set {v, v,..., v n } R m is said to be linearly independent when c v + c v + + c n v n = O only when each c j = 0. Otherwise, the set is said to be linearly dependent. Note that if A = [ ] v v v n is the matrix whose columns are the vj s, then {v, v,..., v n } is linearly independent Ax = O has the trivial solution only. Problem. In each case, determine if the set {v, v,..., v n } is linearly independent or linearly dependent. If the set is linearly dependent, then find a linear dependence relationship among the vectors (that is, find constants c j not all equal to zero so that c v c n v n = O) (a) Let {v, v, v 3 } =,, R (b) Let {v, v, v 3 } = 4, 8, 3 9 R
28 (c) Let {v, v, v 3 } = {[ ] [ [ 0,, R ] 6]}. Property of linear independence. The following theorem gives an important property of linearly independent sets. Theorem (Uniqueness Theorem). If {v, v,..., v n } R m is a linearly independent set and w Span{v, v,..., v n }, then w can be written uniquely as a linear combination of the v i s. That is, we can write w = c v + c v + + c n v n for a unique ordered list of scalars c, c,..., c n. Problem. Demonstrate that the uniqueness theorem is true. 8
29 Facts about linearly independent and linearly dependent sets include the following. If v i = O for some i, then the set {v, v,..., v n } is linearly dependent.. Suppose that v and v are nonzero vectors. Then the set {v, v } is linearly independent if and only if v cv for some c. 3. If m < n, then the set {v, v,..., v n } is linearly dependent. 4. Suppose n and each v i O. Then, the set is linearly dependent if and only if one vector in the set can be written as a linear combination of the others. 5. Let A be the m n matrix whose columns are the v j s: A = [ v v v n ]. Then, the set is linearly independent if and only if the homogeneous system Ax = O has the trivial solution only. 6. Let A be the m n matrix whose columns are the v j s: A = [ v v v n ]. Then, the set is linearly independent if and only if A has a pivot in every column. Problem. Which, if any, of the following sets are linearly independent? Why? 3 0 (a) 0 0, 5 0, 6, (b) 7,, 6, (c) 6, 0, (d) 4. 9
30 .6 Linear and Matrix Transformations.6. Transformations, Images and PreImages. A transformation T : R n R m is a function (or rule) that assigns to each x R n a value T (x) = b R m. Thus, The domain of T is all of nspace: Domain(T ) = R n. The range of T is a subset of mspace: Range(T ) = {T (x) : x R n } R m.. The value T (x) for a given x is also called its image. 3. Each x R n whose image is b R m is said to be a preimage of b. Note that images are unique, but preimages may not be unique. Problem. In each case, find the range of the transformation. That is, find Range(T ). ([ ]) [ ] (a) T : R R x x x with rule T =. x x (b) T : R R with rule T ([ x x ]) [ ] x + 6x =. x 3x 30
31 x (c) T : R 4 R 3 with rule T x x 3 + x 4 x 3 = x + x 4. x x + x + x 3 + x 4 4 (d) T : R 3 R 3 with rule T x x = 3x 3x. x 3 3x 3.6. OneToOne and Onto Transformations. The transformation T : R n R m is said to be onetoone if each b Range(T ) is the image of at most one x R n.. The transformation T : R n R m is said to be onto if Range(T ) = R m. Problem (a), continued. Is T : R R with rule T onto? Explain. ([ x x ]) = [ ] x x x onetoone and/or 3
32 Problem (b), continued. and/or onto? Explain. ([ ]) Is T : R R x with rule T x = [ ] x + 6x x 3x onetoone x Problem (c), continued. Is T : R 4 R 3 with rule T x x 3 + x 4 x 3 = x + x 4 onetoone x x + x + x 3 + x 4 4 and/or onto? Explain. Problem (d), continued. Is T : R 3 R 3 with rule T onto? Explain. x x x 3 = 3x 3x onetoone and/or 3x 3 3
33 .6.3 Linear Transformations T : R n R m is said to be a linear transformation if the following two conditions are satisfied:. T (cv) = ct (v) for c R and v R n, and. T (v + w) = T (v) + T (w) for v, w R n. The conditions for linear transformations lead to the following facts: Fact : If T is a linear transformation and w is a linear combination of v,v,..., v k, then T (w) is a linear combination of T (v ),T (v ),..., T (v k ). In symbols, w = c v + c v + + c k v k = T (w) = c T (v ) + c T (v ) + + c k T (v k ). Fact : If T is a linear transformation, then T maps the span of a set of vectors to the span of the set of images of the vectors. In symbols, ( ) T Span{v, v,..., v k } = Span{T (v ), T (v ),..., T (v k )}. Fact 3: If T (x) = Ax, for some m n matrix A, then T is a linear transformation. Problem. Which, if any, transformations from Problem are linear transformations? Explain. 33
34 Problem 3. Let T : R R be a linear transformation, and suppose that ([ ]) [ ] ([ ]) [ ] 4 3 T = and T =. 4 ([ (a) Find T 6 ]). (b) Find a general formula for T ([ x x ]). 34
35 .6.4 Linear and Matrix Transformations The following theorem says that every linear transformation can be written as a matrix transformation. Formally, Theorem (Matrix Transformations). If T : R n R m is a linear transformation, then the rule for T can be written as T (x) = Ax for a unique m n matrix A. The matrix A, known as the standard matrix of T, has an interesting form. Standard basis vectors and the standard matrix. Let T : R n R m be a linear transformation, and let e j R n be the vector whose j th component is and whose remaining components are 0, for j =,,..., n. Then. Each x R n can be written uniquely as a linear combination of the e j s: 0 0 x x 0 0 x =. = x 0 + x x. n 0 = x. e + x e x n e n... x n 0 0. Since T is a linear transformation, T (x) = x T (e ) + x T (e ) x n T (e n ) [ ] x = T (e ) T (e ) T (e n ) = Ax.. 3. Thus, the rule for T corresponds to matrix multiplication, where A is the matrix whose columns are the images of the e j s: A = [ T (e ) T (e ) T (e n ) ]. x x n The vectors e, e,..., e n are known as the standard basis vectors of R n. If we know the images of the standard basis vectors under the linear transformation T, then we know the standard matrix of T (and the formula for T in terms of the standard matrix). For example, if T : R R 3 is a linear transformation, T (e ) = 3 and T (e ) =, then A =
36 Problem 4. Suppose that T : R R is a linear transformation, [ [ 4 T (e + e ) = and T (e 3] e ) =. 0] Find T (e ), T (e ) and the standard matrix of T. Images of lines in the plane. lines to either lines or points. If T : R R is a linear transformation, then T maps Problem 5. Let T : R R be the transformation with rule [ ] 3 T (x) = x. Find an equation for the image of y = x
37 Problem 6. Let T : R R be the transformation with rule [ ] T (x) = x. 4 Find an equation for the image of y = x (or x = y) Onetoone and onto transformations, revisited. Recall that T : R n R m is said to be onetoone if each b Range(T ) is the image of at most one x R n and that T : R n R m is said to be onto if Range(T ) = R m. The following theorem relates these definitions to properties of the standard matrix of a linear transformation. Theorem (Linear Transformations). Let T : R n R m be the linear transformation with rule T (x) = Ax, where A is an m n matrix. Then. T is onto if and only if the columns of A span all of R m.. T is onetoone if and only if the columns of A form a linearly independent set. Aside : T is onto iff A has a pivot. Aside : T is onetoone iff A has a pivot. 37
38 Problem 7. Which, if any, of the following transformations are onetoone? Which, if any, are onto? Why? ([ ]) x x (a) T : R R 3 x with rule T = 3x x x 3x x (b) T : R 4 R 3 with rule T x x x 3 x 4 x 4x + 8x 3 + x 4 = x 8x 3 + 3x 4 5x 4 (c) T : R 3 R 3 with rule T x x x x = x + 3x + x 3 x 3 3x x + 4x 3 38
39 Footnote: Rotations in space. Recall that positive angles are measured counterclockwise from the positive xaxis, and negative angles are measured clockwise from the positive xaxis. The transformation that rotates points in the plane at an angle θ is a linear transformation. For example, the plot on the right shows the images of the standard basis vectors under the rotation through the positive angle θ = π/ Problem 8. Let T : R R be the rotation about the origin through angle θ. Find the standard matrix of T. Your answer should be in terms of sin(θ) and cos(θ). 39
40 Footnote: Rotations about the zaxis. Assume that positive angles are measured counterclockwise from the positive xaxis, and negative angles are measured clockwise from the positive xaxis, when looking down from the positive zaxis. The transformation that rotates points in 3space at angle θ around the zaxis is a linear transformation. For example, the plot on the right shows the images of the standard basis vectors under the rotation through the positive angle θ = π/5. Problem 9. Let T : R 3 R 3 be the rotation about the zaxis through angle θ. Find the standard matrix of T. Your answer should be in terms of sin(θ) and cos(θ). 40
1 VECTOR SPACES AND SUBSPACES
1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such
More informationDecember 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS
December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in twodimensional space (1) 2x y = 3 describes a line in twodimensional space The coefficients of x and y in the equation
More informationMAT188H1S Lec0101 Burbulla
Winter 206 Linear Transformations A linear transformation T : R m R n is a function that takes vectors in R m to vectors in R n such that and T (u + v) T (u) + T (v) T (k v) k T (v), for all vectors u
More information( ) which must be a vector
MATH 37 Linear Transformations from Rn to Rm Dr. Neal, WKU Let T : R n R m be a function which maps vectors from R n to R m. Then T is called a linear transformation if the following two properties are
More informationSection 9.5: Equations of Lines and Planes
Lines in 3D Space Section 9.5: Equations of Lines and Planes Practice HW from Stewart Textbook (not to hand in) p. 673 # 35 odd, 237 odd, 4, 47 Consider the line L through the point P = ( x, y, ) that
More informationLecture 14: Section 3.3
Lecture 14: Section 3.3 Shuanglin Shao October 23, 2013 Definition. Two nonzero vectors u and v in R n are said to be orthogonal (or perpendicular) if u v = 0. We will also agree that the zero vector in
More informationRecall that two vectors in are perpendicular or orthogonal provided that their dot
Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal
More informationMath 312 Homework 1 Solutions
Math 31 Homework 1 Solutions Last modified: July 15, 01 This homework is due on Thursday, July 1th, 01 at 1:10pm Please turn it in during class, or in my mailbox in the main math office (next to 4W1) Please
More informationDot product and vector projections (Sect. 12.3) There are two main ways to introduce the dot product
Dot product and vector projections (Sect. 12.3) Two definitions for the dot product. Geometric definition of dot product. Orthogonal vectors. Dot product and orthogonal projections. Properties of the dot
More informationLecture notes on linear algebra
Lecture notes on linear algebra David Lerner Department of Mathematics University of Kansas These are notes of a course given in Fall, 2007 and 2008 to the Honors sections of our elementary linear algebra
More information9 Multiplication of Vectors: The Scalar or Dot Product
Arkansas Tech University MATH 934: Calculus III Dr. Marcel B Finan 9 Multiplication of Vectors: The Scalar or Dot Product Up to this point we have defined what vectors are and discussed basic notation
More informationα = u v. In other words, Orthogonal Projection
Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v
More information1 Sets and Set Notation.
LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most
More informationSubspaces of R n LECTURE 7. 1. Subspaces
LECTURE 7 Subspaces of R n Subspaces Definition 7 A subset W of R n is said to be closed under vector addition if for all u, v W, u + v is also in W If rv is in W for all vectors v W and all scalars r
More informationChapter 2: Systems of Linear Equations and Matrices:
At the end of the lesson, you should be able to: Chapter 2: Systems of Linear Equations and Matrices: 2.1: Solutions of Linear Systems by the Echelon Method Define linear systems, unique solution, inconsistent,
More informationJim Lambers MAT 169 Fall Semester 200910 Lecture 25 Notes
Jim Lambers MAT 169 Fall Semester 00910 Lecture 5 Notes These notes correspond to Section 10.5 in the text. Equations of Lines A line can be viewed, conceptually, as the set of all points in space that
More informationSection 13.5 Equations of Lines and Planes
Section 13.5 Equations of Lines and Planes Generalizing Linear Equations One of the main aspects of single variable calculus was approximating graphs of functions by lines  specifically, tangent lines.
More informationEquations Involving Lines and Planes Standard equations for lines in space
Equations Involving Lines and Planes In this section we will collect various important formulas regarding equations of lines and planes in three dimensional space Reminder regarding notation: any quantity
More informationA vector is a directed line segment used to represent a vector quantity.
Chapters and 6 Introduction to Vectors A vector quantity has direction and magnitude. There are many examples of vector quantities in the natural world, such as force, velocity, and acceleration. A vector
More informationMath 215 HW #6 Solutions
Math 5 HW #6 Solutions Problem 34 Show that x y is orthogonal to x + y if and only if x = y Proof First, suppose x y is orthogonal to x + y Then since x, y = y, x In other words, = x y, x + y = (x y) T
More informationProblems. Universidad San Pablo  CEU. Mathematical Fundaments of Biomedical Engineering 1. Author: First Year Biomedical Engineering
Universidad San Pablo  CEU Mathematical Fundaments of Biomedical Engineering Problems Author: First Year Biomedical Engineering Supervisor: Carlos Oscar S. Sorzano June 9, 05 Chapter Lay,.. Solve the
More informationAdding vectors We can do arithmetic with vectors. We ll start with vector addition and related operations. Suppose you have two vectors
1 Chapter 13. VECTORS IN THREE DIMENSIONAL SPACE Let s begin with some names and notation for things: R is the set (collection) of real numbers. We write x R to mean that x is a real number. A real number
More informationSection 1.4. Lines, Planes, and Hyperplanes. The Calculus of Functions of Several Variables
The Calculus of Functions of Several Variables Section 1.4 Lines, Planes, Hyperplanes In this section we will add to our basic geometric understing of R n by studying lines planes. If we do this carefully,
More information521493S Computer Graphics. Exercise 2 & course schedule change
521493S Computer Graphics Exercise 2 & course schedule change Course Schedule Change Lecture from Wednesday 31th of March is moved to Tuesday 30th of March at 1618 in TS128 Question 2.1 Given two nonparallel,
More informationFURTHER VECTORS (MEI)
Mathematics Revision Guides Further Vectors (MEI) (column notation) Page of MK HOME TUITION Mathematics Revision Guides Level: AS / A Level  MEI OCR MEI: C FURTHER VECTORS (MEI) Version : Date: 97 Mathematics
More informationMATH1231 Algebra, 2015 Chapter 7: Linear maps
MATH1231 Algebra, 2015 Chapter 7: Linear maps A/Prof. Daniel Chan School of Mathematics and Statistics University of New South Wales danielc@unsw.edu.au Daniel Chan (UNSW) MATH1231 Algebra 1 / 43 Chapter
More informationMath 241, Exam 1 Information.
Math 241, Exam 1 Information. 9/24/12, LC 310, 11:1512:05. Exam 1 will be based on: Sections 12.112.5, 14.114.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)
More informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationTHREE DIMENSIONAL GEOMETRY
Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,
More informationLinear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices
MATH 30 Differential Equations Spring 006 Linear algebra and the geometry of quadratic equations Similarity transformations and orthogonal matrices First, some things to recall from linear algebra Two
More informationISOMETRIES OF R n KEITH CONRAD
ISOMETRIES OF R n KEITH CONRAD 1. Introduction An isometry of R n is a function h: R n R n that preserves the distance between vectors: h(v) h(w) = v w for all v and w in R n, where (x 1,..., x n ) = x
More informationSolutions to old Exam 1 problems
Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections
More informationOrthogonal Projections
Orthogonal Projections and Reflections (with exercises) by D. Klain Version.. Corrections and comments are welcome! Orthogonal Projections Let X,..., X k be a family of linearly independent (column) vectors
More information11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space
11 Vectors and the Geometry of Space 11.1 Vectors in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. 2 Objectives! Write the component form of
More informationInner Product Spaces and Orthogonality
Inner Product Spaces and Orthogonality week 34 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,
More informationSolving Simultaneous Equations and Matrices
Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering
More informationLinear Algebra Done Wrong. Sergei Treil. Department of Mathematics, Brown University
Linear Algebra Done Wrong Sergei Treil Department of Mathematics, Brown University Copyright c Sergei Treil, 2004, 2009, 2011, 2014 Preface The title of the book sounds a bit mysterious. Why should anyone
More information1.5 Equations of Lines and Planes in 3D
40 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE Figure 1.16: Line through P 0 parallel to v 1.5 Equations of Lines and Planes in 3D Recall that given a point P = (a, b, c), one can draw a vector from
More informationLINEAR ALGEBRA W W L CHEN
LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,
More informationPrerequisites: TSI Math Complete and high school Algebra II and geometry or MATH 0303.
Course Syllabus Math 1314 College Algebra Revision Date: 82115 Catalog Description: Indepth study and applications of polynomial, rational, radical, exponential and logarithmic functions, and systems
More informationVector Spaces; the Space R n
Vector Spaces; the Space R n Vector Spaces A vector space (over the real numbers) is a set V of mathematical entities, called vectors, U, V, W, etc, in which an addition operation + is defined and in which
More informationOPRE 6201 : 2. Simplex Method
OPRE 6201 : 2. Simplex Method 1 The Graphical Method: An Example Consider the following linear program: Max 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2
More information12.5 Equations of Lines and Planes
Instructor: Longfei Li Math 43 Lecture Notes.5 Equations of Lines and Planes What do we need to determine a line? D: a point on the line: P 0 (x 0, y 0 ) direction (slope): k 3D: a point on the line: P
More informationDirect Methods for Solving Linear Systems. Matrix Factorization
Direct Methods for Solving Linear Systems Matrix Factorization Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University c 2011
More informationJUST THE MATHS UNIT NUMBER 8.5. VECTORS 5 (Vector equations of straight lines) A.J.Hobson
JUST THE MATHS UNIT NUMBER 8.5 VECTORS 5 (Vector equations of straight lines) by A.J.Hobson 8.5.1 Introduction 8.5. The straight line passing through a given point and parallel to a given vector 8.5.3
More informationCS3220 Lecture Notes: QR factorization and orthogonal transformations
CS3220 Lecture Notes: QR factorization and orthogonal transformations Steve Marschner Cornell University 11 March 2009 In this lecture I ll talk about orthogonal matrices and their properties, discuss
More informationWhat are the place values to the left of the decimal point and their associated powers of ten?
The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything
More informationLinear Algebra Done Wrong. Sergei Treil. Department of Mathematics, Brown University
Linear Algebra Done Wrong Sergei Treil Department of Mathematics, Brown University Copyright c Sergei Treil, 2004, 2009, 2011, 2014 Preface The title of the book sounds a bit mysterious. Why should anyone
More informationVectors 2. The METRIC Project, Imperial College. Imperial College of Science Technology and Medicine, 1996.
Vectors 2 The METRIC Project, Imperial College. Imperial College of Science Technology and Medicine, 1996. Launch Mathematica. Type
More informationAlgebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 201213 school year.
This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra
More informationCollege Algebra. Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGrawHill, 2008, ISBN: 9780072867381
College Algebra Course Text Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGrawHill, 2008, ISBN: 9780072867381 Course Description This course provides
More informationi=(1,0), j=(0,1) in R 2 i=(1,0,0), j=(0,1,0), k=(0,0,1) in R 3 e 1 =(1,0,..,0), e 2 =(0,1,,0),,e n =(0,0,,1) in R n.
Length, norm, magnitude of a vector v=(v 1,,v n ) is v = (v 12 +v 22 + +v n2 ) 1/2. Examples v=(1,1,,1) v =n 1/2. Unit vectors u=v/ v corresponds to directions. Standard unit vectors i=(1,0), j=(0,1) in
More informationSection 11.4: Equations of Lines and Planes
Section 11.4: Equations of Lines and Planes Definition: The line containing the point ( 0, 0, 0 ) and parallel to the vector v = A, B, C has parametric equations = 0 + At, = 0 + Bt, = 0 + Ct, where t R
More informationLINEAR ALGEBRA W W L CHEN
LINEAR ALGEBRA W W L CHEN c W W L Chen, 1982, 2008. This chapter originates from material used by author at Imperial College, University of London, between 1981 and 1990. It is available free to all individuals,
More information1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two nonzero vectors u and v,
1.3. DOT PRODUCT 19 1.3 Dot Product 1.3.1 Definitions and Properties The dot product is the first way to multiply two vectors. The definition we will give below may appear arbitrary. But it is not. It
More informationTwo vectors are equal if they have the same length and direction. They do not
Vectors define vectors Some physical quantities, such as temperature, length, and mass, can be specified by a single number called a scalar. Other physical quantities, such as force and velocity, must
More informationLINES AND PLANES CHRIS JOHNSON
LINES AND PLANES CHRIS JOHNSON Abstract. In this lecture we derive the equations for lines and planes living in 3space, as well as define the angle between two nonparallel planes, and determine the distance
More informationNotes from February 11
Notes from February 11 Math 130 Course web site: www.courses.fas.harvard.edu/5811 Two lemmas Before proving the theorem which was stated at the end of class on February 8, we begin with two lemmas. The
More informationLINES AND PLANES IN R 3
LINES AND PLANES IN R 3 In this handout we will summarize the properties of the dot product and cross product and use them to present arious descriptions of lines and planes in three dimensional space.
More informationGCE Mathematics (6360) Further Pure unit 4 (MFP4) Textbook
Version 36 klm GCE Mathematics (636) Further Pure unit 4 (MFP4) Textbook The Assessment and Qualifications Alliance (AQA) is a company limited by guarantee registered in England and Wales 364473 and a
More information... ... . (2,4,5).. ...
12 Three Dimensions ½¾º½ Ì ÓÓÖ Ò Ø ËÝ Ø Ñ So far wehave been investigatingfunctions ofthe form y = f(x), withone independent and one dependent variable Such functions can be represented in two dimensions,
More information1 Introduction. Linear Programming. Questions. A general optimization problem is of the form: choose x to. max f(x) subject to x S. where.
Introduction Linear Programming Neil Laws TT 00 A general optimization problem is of the form: choose x to maximise f(x) subject to x S where x = (x,..., x n ) T, f : R n R is the objective function, S
More informationdiscuss how to describe points, lines and planes in 3 space.
Chapter 2 3 Space: lines and planes In this chapter we discuss how to describe points, lines and planes in 3 space. introduce the language of vectors. discuss various matters concerning the relative position
More informationEquations of Lines and Planes
Calculus 3 Lia Vas Equations of Lines and Planes Planes. A plane is uniquely determined by a point in it and a vector perpendicular to it. An equation of the plane passing the point (x 0, y 0, z 0 ) perpendicular
More information3. INNER PRODUCT SPACES
. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.
More informationis in plane V. However, it may be more convenient to introduce a plane coordinate system in V.
.4 COORDINATES EXAMPLE Let V be the plane in R with equation x +2x 2 +x 0, a twodimensional subspace of R. We can describe a vector in this plane by its spatial (D)coordinates; for example, vector x 5
More informationSolutions for Review Problems
olutions for Review Problems 1. Let be the triangle with vertices A (,, ), B (4,, 1) and C (,, 1). (a) Find the cosine of the angle BAC at vertex A. (b) Find the area of the triangle ABC. (c) Find a vector
More informationCopyrighted Material. Chapter 1 DEGREE OF A CURVE
Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two
More informationVector Spaces. Chapter 2. 2.1 R 2 through R n
Chapter 2 Vector Spaces One of my favorite dictionaries (the one from Oxford) defines a vector as A quantity having direction as well as magnitude, denoted by a line drawn from its original to its final
More informationRecall the basic property of the transpose (for any A): v A t Aw = v w, v, w R n.
ORTHOGONAL MATRICES Informally, an orthogonal n n matrix is the ndimensional analogue of the rotation matrices R θ in R 2. When does a linear transformation of R 3 (or R n ) deserve to be called a rotation?
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More informationA Concrete Introduction. to the Abstract Concepts. of Integers and Algebra using Algebra Tiles
A Concrete Introduction to the Abstract Concepts of Integers and Algebra using Algebra Tiles Table of Contents Introduction... 1 page Integers 1: Introduction to Integers... 3 2: Working with Algebra Tiles...
More informationAlgebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
More informationYou know from calculus that functions play a fundamental role in mathematics.
CHPTER 12 Functions You know from calculus that functions play a fundamental role in mathematics. You likely view a function as a kind of formula that describes a relationship between two (or more) quantities.
More informationx1 x 2 x 3 y 1 y 2 y 3 x 1 y 2 x 2 y 1 0.
Cross product 1 Chapter 7 Cross product We are getting ready to study integration in several variables. Until now we have been doing only differential calculus. One outcome of this study will be our ability
More information3 Orthogonal Vectors and Matrices
3 Orthogonal Vectors and Matrices The linear algebra portion of this course focuses on three matrix factorizations: QR factorization, singular valued decomposition (SVD), and LU factorization The first
More informationState of Stress at Point
State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,
More informationVELOCITY, ACCELERATION, FORCE
VELOCITY, ACCELERATION, FORCE velocity Velocity v is a vector, with units of meters per second ( m s ). Velocity indicates the rate of change of the object s position ( r ); i.e., velocity tells you how
More information26. Determinants I. 1. Prehistory
26. Determinants I 26.1 Prehistory 26.2 Definitions 26.3 Uniqueness and other properties 26.4 Existence Both as a careful review of a more pedestrian viewpoint, and as a transition to a coordinateindependent
More informationOrthogonal Projections and Orthonormal Bases
CS 3, HANDOUT A, 3 November 04 (adjusted on 7 November 04) Orthogonal Projections and Orthonormal Bases (continuation of Handout 07 of 6 September 04) Definition (Orthogonality, length, unit vectors).
More informationDiscrete Convolution and the Discrete Fourier Transform
Discrete Convolution and the Discrete Fourier Transform Discrete Convolution First of all we need to introduce what we might call the wraparound convention Because the complex numbers w j e i πj N have
More informationSection 11.1: Vectors in the Plane. Suggested Problems: 1, 5, 9, 17, 23, 2537, 40, 42, 44, 45, 47, 50
Section 11.1: Vectors in the Plane Page 779 Suggested Problems: 1, 5, 9, 17, 3, 537, 40, 4, 44, 45, 47, 50 Determine whether the following vectors a and b are perpendicular. 5) a = 6, 0, b = 0, 7 Recall
More informationPythagorean vectors and their companions. Lattice Cubes
Lattice Cubes Richard Parris Richard Parris (rparris@exeter.edu) received his mathematics degrees from Tufts University (B.A.) and Princeton University (Ph.D.). For more than three decades, he has lived
More informationIntroduction Assignment
PRECALCULUS 11 Introduction Assignment Welcome to PREC 11! This assignment will help you review some topics from a previous math course and introduce you to some of the topics that you ll be studying
More informationPRIMARY CONTENT MODULE Algebra I Linear Equations & Inequalities T71. Applications. F = mc + b.
PRIMARY CONTENT MODULE Algebra I Linear Equations & Inequalities T71 Applications The formula y = mx + b sometimes appears with different symbols. For example, instead of x, we could use the letter C.
More informationUndergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics
Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics A Semester Course in Finite Mathematics for Business and Economics Marcel B. Finan c All Rights Reserved August 10,
More informationSection 2.4: Equations of Lines and Planes
Section.4: Equations of Lines and Planes An equation of three variable F (x, y, z) 0 is called an equation of a surface S if For instance, (x 1, y 1, z 1 ) S if and only if F (x 1, y 1, z 1 ) 0. x + y
More informationVISUAL ALGEBRA FOR COLLEGE STUDENTS. Laurie J. Burton Western Oregon University
VISUAL ALGEBRA FOR COLLEGE STUDENTS Laurie J. Burton Western Oregon University VISUAL ALGEBRA FOR COLLEGE STUDENTS TABLE OF CONTENTS Welcome and Introduction 1 Chapter 1: INTEGERS AND INTEGER OPERATIONS
More information1. Vectors and Matrices
E. 8.02 Exercises. Vectors and Matrices A. Vectors Definition. A direction is just a unit vector. The direction of A is defined by dir A = A, (A 0); A it is the unit vector lying along A and pointed like
More informationMathematics 205 HWK 6 Solutions Section 13.3 p627. Note: Remember that boldface is being used here, rather than overhead arrows, to indicate vectors.
Mathematics 205 HWK 6 Solutions Section 13.3 p627 Note: Remember that boldface is being used here, rather than overhead arrows, to indicate vectors. Problem 5, 13.3, p627. Given a = 2j + k or a = (0,2,
More informationContent. Chapter 4 Functions 61 4.1 Basic concepts on real functions 62. Credits 11
Content Credits 11 Chapter 1 Arithmetic Refresher 13 1.1 Algebra 14 Real Numbers 14 Real Polynomials 19 1.2 Equations in one variable 21 Linear Equations 21 Quadratic Equations 22 1.3 Exercises 28 Chapter
More informationMath 241 Lines and Planes (Solutions) x = 3 3t. z = 1 t. x = 5 + t. z = 7 + 3t
Math 241 Lines and Planes (Solutions) The equations for planes P 1, P 2 and P are P 1 : x 2y + z = 7 P 2 : x 4y + 5z = 6 P : (x 5) 2(y 6) + (z 7) = 0 The equations for lines L 1, L 2, L, L 4 and L 5 are
More informationMathematics Review for MS Finance Students
Mathematics Review for MS Finance Students Anthony M. Marino Department of Finance and Business Economics Marshall School of Business Lecture 1: Introductory Material Sets The Real Number System Functions,
More informationUnderstanding Basic Calculus
Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other
More informationSECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS
SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS Assume f ( x) is a nonconstant polynomial with real coefficients written in standard form. PART A: TECHNIQUES WE HAVE ALREADY SEEN Refer to: Notes 1.31
More information8. Linear leastsquares
8. Linear leastsquares EE13 (Fall 21112) definition examples and applications solution of a leastsquares problem, normal equations 81 Definition overdetermined linear equations if b range(a), cannot
More informationExample SECTION 131. XAXIS  the horizontal number line. YAXIS  the vertical number line ORIGIN  the point where the xaxis and yaxis cross
CHAPTER 13 SECTION 131 Geometry and Algebra The Distance Formula COORDINATE PLANE consists of two perpendicular number lines, dividing the plane into four regions called quadrants XAXIS  the horizontal
More informationLines and Planes 1. x(t) = at + b y(t) = ct + d
1 Lines in the Plane Lines and Planes 1 Ever line of points L in R 2 can be epressed as the solution set for an equation of the form A + B = C. The equation is not unique for if we multipl both sides b
More informationCourse Outlines. 1. Name of the Course: Algebra I (Standard, College Prep, Honors) Course Description: ALGEBRA I STANDARD (1 Credit)
Course Outlines 1. Name of the Course: Algebra I (Standard, College Prep, Honors) Course Description: ALGEBRA I STANDARD (1 Credit) This course will cover Algebra I concepts such as algebra as a language,
More information