MATH 2030: SYSTEMS OF LINEAR EQUATIONS. ax + by + cz = d. )z = e. while these equations are not linear: xy z = 2, x x = 0,


 Andrea Craig
 11 months ago
 Views:
Transcription
1 MATH 23: SYSTEMS OF LINEAR EQUATIONS Systems of Linear Equations In the plane R 2 the general form of the equation of a line is ax + by = c and that the general equation of a plane in R 3 will be we call these type of equations linear. ax + by + cz = d Definition.. A linear equation in the n variables x, x 2,..., x n is an equation that may be written in the form a x + b 2 x a n x n = b where the coefficients a,..., a n and the constant term are constants. Example.2. The following equations are linear: 2x + y = 3, r 2 3 s 5 8 t = 3, x x 2 + 2x 3 = 3, 2x + π 4 y tan( 4 2 )z = e. while these equations are not linear: xy z = 2, x 2 + x 2 2 =, x y = 3z, 2r + s =, 2x + π 2 4 sin( 4 2z) = e. A solution of a linear equation a x +...+a n x n = b is a vector [s, s 2,..., s n ] whose components satsify the equation when we substitute x i = s i for i [, n]. Example.3. In the first example, one possible solution to the first linear equation would be [2, ] since the substitution of x = 2 and y = yields 2(2) + ( ) = 3. The vector [, ] is another solution. We already know this describes a line in the plane, and this may be written parametrically by letting x = t and solving for y, to produce [t, 3 2t]. The linear equation x x 2 + 2x 3 = 3 has [3,, ] and [,, 2] and [6,, ] as specific solutions. Of course, in R 3 this describes a plane. To see this, set x 2 = s and x 3 = t then the solutions are described parametrically by [3 + s 2t, s, t]. A system of linear equations is a fine set of linear equation, each with the same variables. A solution of a system of linear equations is a vector that is simultaneously a solution of each equation in the system. The solution set of a system of linear equations will be the set of all solutions of the system. We will call the process of calculating the solution set, solving the system. Example.4. The system, x + y =, x y = has [ 2, 2 ] as a solution, since it satisfies both equations. Notice that the vector [, ] is not a solution as it only satisfies the second equation and not the first. Example.5. In R 2 there are three typical cases for linear equations x + y =, x y =, here the lines intersect once with the solution [ 2, 2 ].
2 2 MATH 23: SYSTEMS OF LINEAR EQUATIONS x y =, 2x 2y = 2, here the lines intersect infinitely many times with the form [t, t ]. x y =, x y = 2, this equation has no solution. Geometrically these represent parallel lines, to see that this has no solution solve for in the first and substitute into the second, we find that = 2 which cannot happen on the real line. From these examples we add a few terms: a system of linear equations is consistent if it has at least one solution, a system with no solutions will be called inconsistent. Despite their simplicity the three systems in the previous example illustrate the only three possibilities for the number of solutions of a system of linear equations with real coefficients. For the moment this will be unproven but we have the following Proposition for any system of linear equations Proposition.6. A system of linear equations with real coefficients has either A unique solution (consistent) infinitely many solutions (consistent) no solutions (inconsistent). Solving a System of Linear Equations. To start we must introduce the concept of equivalence between linear systems, we say two linear systems are equivalent if they have the same solution sets. As an example, x y =, x + y = 3, x y =, y = have the same unique solution [2, ] and so they are equivalent. This example illustrates how we will go about finding the solution to a given system of linear equations; by finding increasingly simpler yet equivalent linear systems we may determine the solution by inspection, as in this example where the triangular pattern of the second system gives y = automatically. Example.7. Q What is the solution to the system x y z = 2, y + 3z = 5, 5z = A: Taking the last equation, solving for z = 2 and substituting this value into the remaining two we find that these become: x y = 4, y =. Repeating the process with y = we find x = 3. Thus [3,, 2] is a solution to this system. In this last example we used back substitution, using this tool we will determine a general strategy for transforming a given system into an equivalent one that can be solved easily. Without saying too much we will illustrate this approach with one last example Example.8. Q: Solve the system x y z = 2, 3x 3y + 2z = 6, 2x y + z = 9. A: As a start towards reducing this to a simple triangular form, as in the last example, we would like to remove the coefficient of the xterm to zero in the second and third equation. This may be done by multiplying the first equation by an appropriate constant and subtracting this new equation from the one we wish to
3 MATH 23: SYSTEMS OF LINEAR EQUATIONS 3 change. This will not affect x,y or z and so this may be written more compactly as x y = 6, or z where the first three columns contain the coefficients of the variables in order, the final column contains the constant terms and the vertical bar is meant to differentiate equality versus a sum of terms. This matrix is called the augmented matrix of the system. As we have yet to introduce the operation that will act on the matrix we will work with the system as linear equations and illustrate how our actions reflect on the augmented matrix of each system. x y z = 2 3x 3y + 2z = 6 2x y + z = Subtract 3 times the first equation from the second equation x y z = 2 2 5z = 5 2x y + z = Subtract 2 times the first equation from the third equation, x y z = 2 2 5z = 5 y + 3 = Interchange the second and third equations x y z = 2 2 y + 3 = z = 5 We can stop here, we ve found a simpler equivalent system from which we can read off the required values for this system as [3,, 2]. Thus these two examples are equivalent systems despite appearances otherwise. The above calculation shows that any solution of the given system is also a solution of the final one. As this process is reversible we have found a way to change from one equivalent system to another and backwards if need be. Furthermore these operations may be expressed as operations on the components of matrices so we may as well work with matrices since they are equivalent to systems of linear equations. Direct Methods for Solving Linear Systems We want to make this procedure more systematic and generalized for any system of linear equations. We will do this by reducing the augmented matrix of a system of linear equations to a simpler form where back substitution produces the solution. Echelon form of matrices. For any linear system we will define two helpful matrices that will be important in the work to come, the coefficient matrix containing the coefficients of the variables, and the augmented matrix which is the coefficient matrix with an extra column added containing the constant terms of the linear system.
4 4 MATH 23: SYSTEMS OF LINEAR EQUATIONS From the last example, the augmented matrix related to the linear system is again, x y z = 2 2 3x 3y + 2z = x y + z = and so the coefficient matrix for this system is If a variable, say x i is missing in the j th linear equation in the system, the (i, j)th component will have a zero. Symbolically if we denote the coefficient matrix as A and the column vector of the constant term for each equation as b, the augmented matrix is [A b]. There will be times where a matrix may not be simplified to a triangular form; it is possible to simplify a matrix to a another helpful form nonetheless Definition.9. A matrix is in row echelon form if it satisfies the properties Any rows consisting entirely of zeros are at the bottom. In each nonzero row, the first nonzero entry, the leading entry is in a column to the left of any leading entries below it. Example.. All of these matrices are in echelon form: , 2 3, For any linear system, the equivalent augmented matrix which has been reduced to row echelon form may be solved using back substitution. Example.. Q: If we assume that each of the above matrices are augmented matrices, determine their solutions if any. A: Transforming each augmented matrix to the corresponding set of linear equations will facilitate back substitution () The equations are x + y = 2, 3y = 5 and so the solution will be [ 3, 5 3 ]. (2) Here the linear system is x = 3, 2x 2 = 3 and = 2. This cannot happen and so the system has no solution. (3) Noticing that the bottom row of this matrix implies = 5 and so we conclude no solution exists. Elementary Row Operations. To describe the procedure for reducing a matrix to row echelon form we should define the operations on a matrix that maintain the equivalence between linear systems and their augmented matrices, called elementary row operations. Definition.2. The following elementary operations can be performed on a matrix () Interchange two rows. (2) Multiply a row by a nonzero constant. (3) Add a multiple of a row to another row.
5 MATH 23: SYSTEMS OF LINEAR EQUATIONS 5 To facilitate calculations we will use a short hand to denote the row operations symbolically R i R j denotes an interchange of the ith and jth rows. kr i implies we multiply the ith row by k. Adding a multiple of the ith row to the jth row, is simply kr i + R j The third rule allows one to subtract a row from another using k = and that division of a row s constants by a number r is easily done using k = r and the second rule. The process of applying elementary row operations to bring a matrix into row echelon form is called row reduction. Example.3. Q: Reduce the matrix to echelon form: A: We are going to work from the top left to the bottom right. The idea will be to work with the leading entry in the topmost row and use it to create zeros below it, we call this a pivot and this subprocess is called pivoting, typically one uses row operations to set the pivot to equal one as well. With this in mind, we eliminate the entries below the in the toprightmost corner Composing the row operations R 2 + 2R and R 3 3R yields The first column is in echelon form, we now move onto the next column. To simplify this matrix we interchange the rows R 2 R 3 and scale the new R 2 by We have found the second pivot, we must eliminate the seven below it to have this column in echelon form, via R 3 7R 2, At this point the entire matrix has been reduced to row echelon form. Elementary row operations are reversible, so that if a matrix A is transformed to a new matrix B by any combination of operations, there is a corresponding inverse operation to transform B into A. Definition.4. Matrices A and B are equivalent if there is a sequence of elementary row operations that converts A into B. The matrices in the previous example are equivalent. To generalize this idea we return to row echelon form. Theorem.5. Matrices A and B are equivalent if and only if they can be reduced to the same row echelon form.
6 6 MATH 23: SYSTEMS OF LINEAR EQUATIONS Proof. Supposing A and B are row equivalent, by composing the combination of row operation that converts A into B to the combination converting B into the row echelon form E, we have constructed a combination of row operations that take A to E. To prove the other direction, we suppose both A and B both have a different combination of row operations that bring them into the same row echelon form E. Inverting the combinations of elementary row operations that take B to E and appending this to the list of row operations that take A to E, we find a set of row operations that take A to B. Gaussian Elimination. The process of using elementary row operations to alter the augmented matrix of a system of linear equations we produce an equivalent system that is more easily solved using back substitution. More formally this procedure is called Gaussian elimination and it consists of the three steps () Find the augmented matrix of the system of linear equations (2) Use elementary row operations to reduce the augmented matrix to row echelon form. (3) Using back substitution, solve the equivalent system that corresponds to the rowreduced matrix. Example.6. Q: Solve the system 2x 2 + 3x 3 = 8 2x + 3x 2 + x 3 = 5, x x 2 2x 3 = 5. A: The augmented matrix is As there is a one in the first column of the third row we swap it for the first row, R R 3 and then eliminate the 2 in the second row directly below the new pivot, R 2 2R, Scaling the second row by 5 and following this by R 3 2R 2 produces the matrix The equivalent system of linear equations that correspond to this augmented matrix will be x x 2 2x 3 = 5, x 2 + x 3 = 3, x 3 = 2; back substitution gives the solution = 3 x x 2 x 3
7 MATH 23: SYSTEMS OF LINEAR EQUATIONS 7 Example.7. Q: Solve the system w 2x + y z = 2 w x + y 2z = 5 2w + 2x 2y + 4z = A: The augmented matrix will be As the first row has the leading coefficient as, we will use this as the first pivot. Applying the elementary row operation R 2 R and R 3 + 2R we produce the matrix One more row operation R 3 + 2R 2 will put this into row echelon form The linear system associated with this augmented matrix is w 2x + y z = 2, x z = 3 this will have infinitely many solutions, if we express the variables related to the leading entries of the matrix (the leading variables) in terms of the other variables (the free variables ). Substituting x = z + 3 into the equation for w, if we treat the free variables as parameters, the solution may be expressed in vector form as w 8 3 x y = 3 + y + z z This second example will showcase the fact that the free variables are just the variables that aren t leading variables. The number of leading coefficients is the number of nonzero rows in the row echelon form of the coefficient matrix, we will be able to predict the number of free variables before the solution is determined using back substitution. Definition.8. The rank of a matrix is the number of nonzero rows in its row echelon form. We will call the rank of a matrix A by rank(a), in the first example, the rank was 3, and in the second example the rank will be two (by construction). Theorem.9. Rank Theorem: Let A be the coefficient matrix of a system of linear equations with n variables. If the system is consistent, then: the number of free variables = n  rank(a).
8 8 MATH 23: SYSTEMS OF LINEAR EQUATIONS Example.2. Q: Solve the system A: The augmented matrix is now x x 2 + 2x 3 = 3 x + 2x 2 x 3 = 3 2x 2 2x 3 = subtracting the first row from the second, and scaling this by three we find To eliminate the column entries below the leading entry in the second row, R 3 2R 2 gives this matrix is inconsistent as it is giving a clear contradiction = 5, it has no solutions. GaussJordan Elimination. We modify the Gaussian elimination so that back substitution is easily still. This will be helpful for when calculations are being done by hand on a system with infinitely many solutions, this is done by changing the row echelon matrix further. Definition.2. A matrix is in reduced row echelon form if it satisfies the following properties () It is already in row echelon form. (2) The leading entry in each nonzero row is a (called a leading ). (3) Each column containing a leading has zeros everywhere else. As an example here are any of the 2 2 matrices in reduced row echelon form, [ ] [ ] [ ] [ ] a,,,. where a is any number in R. Unlike row echelon form, the reduced row echelon form is unique. That is for each matrix A there is only one matrix R in reduced row echelon form that is equivalent to A. As with Gaussian elimination we introduce GaussJordan elimination whose steps consist of: () Write the augmented matrix of the system of linear equations. (2) Use elementary row operations to reduce the augmented matrix to a reduce row echelon form. (3) If the resulting system is consistent, solve for the leading variables in terms of the remaining free variables (if any).,
9 MATH 23: SYSTEMS OF LINEAR EQUATIONS 9 Example.22. Q: Apply the GaussJordan elimination algorithm on the example in (.7) A: We already know this linear system has an augmented matrix which is equivalent to to put this into reduced row echelon form, apply the row operation R + 2R 2 to get Immediately we find the solutions are of the form w 8 3 x y = 3 + y + z z Example.23. Q:Consider the line of intersection of the two planes x 2y 2z = 7 x + 3y + 4z = 2 A: The augmented matrix will be [ then by adding the first row from the second brings the metric into row echelon form (R 2 + R ). [ ] One more row operation brings this into reduced echelon form R + 2R 2 [ ] with the associated linear system: x + 2z = 3, y + 2z = 5 Choosing x and y as our leading variables and z as the free variable the equation of the line may be written in vector form x 3 2 y = 5 + z 2 z 3 Example.24. Q: Let p =, q = 2, u = and v =. Determine whether the lines x = p + tu and x = q + sv intersect, and if so, where. ]
10 MATH 23: SYSTEMS OF LINEAR EQUATIONS A: If these lines intersect, there should be a solution x = x y that satisfies both z equations at once. i.e., p+tu = x = q+sv and so p+tu = q+sv or tu sv = q p. Using the parametric form for these lines, we find t 3s =, s + t = 2, s + t = 2 the solution is easily checked to be t = 5 4 and s = 3 4. Supposing x = p + tu we find that x =. Homogeneous Systems. So far we have argued that every system of linear equations has either no solution, a unique solution or infinitely many solutions. There is a special type of system that always has at least one solution. Definition.25. A system of linear equations is called homogeneous if the constant term in each equation is zero. Symbolically, this means the augmented matrix is of the form [A ]; every linear system has an associated homogeneous linear system produced by replacing the b vector with the zero vector. These systems will never be inconsistent and so it must have either infinitely many solutions or a unique one. In the first case we have a helpful theorem to determine when a solution has infinitely many solutions, Theorem.26. If [A ] is a homogeneous system of m linear equations with n variables, where m < n, then the system has infinitely many solutions. Proof. At the very least x = will be a solution, it will be consistent. By the Rank theorem we know that the rank(a) m, as these may be seen as the number of nontrivial linear equations recorded as rows in the matrix it must be less than the number of linear equations in the system. Furthermore we see that the number of free variables will equal n rank(a) n m >, there will be at least one free variable (as m and n are integers) hence there will be infinitely many solutions. Linear Systems over Z p. We have only considered linear systems with real valued coefficients, which lead to vector solutions in R n. Returning to the idea of a code vector, we ask what the solutions of linear equations with coefficients in Z p. When p is a prime number Z p behaves like R in many ways  one can add, subtract, multiply and divide numbers in a way that is reversible as well. This is the important part, because it allows us to solve systems of linear equations when the variables and coefficients belong to Z p for some prime p, this is called solving the system over Z p. Consider the example x + x 2 + x 3 = with coefficients in Z 2, it will have four solutions due to the finite nature of the field Z 2 : x x 2 x 3 =,,,
11 MATH 23: SYSTEMS OF LINEAR EQUATIONS In Z 3 we will have quite a few more, by considering the ways we may add a triplet with values in Z 2 to sum to 3: 3 : + + = mod3 3 : = mod3 3 : = mod3 thus there will be nine possible solutions in Z 3. We will not need to do such combinatoric guesswork, as in R n, GaussJordan elimination will work here as well. Example.27. Q:Solve the following linear system of linear equations over Z 3, x + 2x 2 + x 3 = x + x 3 = 2 x 2 + 2x 3 = A: In Z 2, = 2 mod3, and so subtraction will not be needed, similarly division is unnecessary as 2 2 = mod3. The augmented matrix is To begin row reduction, apply R R, following this with R + R 2 and R 3 + 2R 2 we find This matrix is now in row echelon form, to simplify to reduced row echelon form apply the row operations R + R 2 and 2R 3, 2 Example.28. Q: Solve the system of linear equations over Z 2, x + x 2 + x 3 + x 4 = x + x 2 = x 2 + x 3 = x 3 + x 4 = x + x 4 =
12 2 MATH 23: SYSTEMS OF LINEAR EQUATIONS A: To start we write the augmented matrix As the leading entry of the first column is a, we may use this as a pivot, and so to remove the remaining nonzero components in this column we apply the row operations R 2 + R and R 5 + R as these are the only rows containing nonzero entries As the third row has a nonzero entry in the second column, we swap this with the second row (R 3 R 2 ) and use it as the next pivot. Applying R + R 2 and R 5 + R 2 to eliminate the remaining nonzero entries in this column: Here the third pivot appears in the third row, to put this into reduced row echelon form apply the row operations R 2 + R 3 and R 4 + R 3 : Writing down the associated linear system, this becomes x + x 4 =, x 2 + x 4 = and x 3 + x 4 =, the leading variables are x, x 2 and x 3 and x 4 is a free variable. In vector form this becomes x x 2 x 3 x 4 = + x 4 as x 4 =, we see the only vectors that are solutions will be s t = {[,,, ], [,,, ]}. References [] D. Poole, Linear Algebra: A modern introduction  3rd Edition, Brooks/Cole (22).
MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An ndimensional vector is a row or a column of n numbers (or letters): a 1.
MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0534405967. Systems of Linear Equations Definition. An ndimensional vector is a row or a column
More information1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form
1. LINEAR EQUATIONS A linear equation in n unknowns x 1, x 2,, x n is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b, where a 1, a 2,..., a n, b are given real numbers. For example, with x and
More informationa 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given
More information1.2 Solving a System of Linear Equations
1.. SOLVING A SYSTEM OF LINEAR EQUATIONS 1. Solving a System of Linear Equations 1..1 Simple Systems  Basic De nitions As noticed above, the general form of a linear system of m equations in n variables
More informationSolving Systems of Linear Equations Using Matrices
Solving Systems of Linear Equations Using Matrices What is a Matrix? A matrix is a compact grid or array of numbers. It can be created from a system of equations and used to solve the system of equations.
More informationSolving Systems of Linear Equations
LECTURE 5 Solving Systems of Linear Equations Recall that we introduced the notion of matrices as a way of standardizing the expression of systems of linear equations In today s lecture I shall show how
More informationSolutions to Linear Algebra Practice Problems 1. form (because the leading 1 in the third row is not to the right of the
Solutions to Linear Algebra Practice Problems. Determine which of the following augmented matrices are in row echelon from, row reduced echelon form or neither. Also determine which variables are free
More informationSystems of Linear Equations
Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in 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
More informationSolving Systems of Linear Equations; Row Reduction
Harvey Mudd College Math Tutorial: Solving Systems of Linear Equations; Row Reduction Systems of linear equations arise in all sorts of applications in many different fields of study The method reviewed
More informationMath 240: Linear Systems and Rank of a Matrix
Math 240: Linear Systems and Rank of a Matrix Ryan Blair University of Pennsylvania Thursday January 20, 2011 Ryan Blair (U Penn) Math 240: Linear Systems and Rank of a Matrix Thursday January 20, 2011
More informationSolutions to Math 51 First Exam January 29, 2015
Solutions to Math 5 First Exam January 29, 25. ( points) (a) Complete the following sentence: A set of vectors {v,..., v k } is defined to be linearly dependent if (2 points) there exist c,... c k R, not
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 informationRow Echelon Form and Reduced Row Echelon Form
These notes closely follow the presentation of the material given in David C Lay s textbook Linear Algebra and its Applications (3rd edition) These notes are intended primarily for inclass presentation
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationLinear Dependence Tests
Linear Dependence Tests The book omits a few key tests for checking the linear dependence of vectors. These short notes discuss these tests, as well as the reasoning behind them. Our first test checks
More informationDiagonal, Symmetric and Triangular Matrices
Contents 1 Diagonal, Symmetric Triangular Matrices 2 Diagonal Matrices 2.1 Products, Powers Inverses of Diagonal Matrices 2.1.1 Theorem (Powers of Matrices) 2.2 Multiplying Matrices on the Left Right by
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
More informationSolving Linear Systems, Continued and The Inverse of a Matrix
, Continued and The of a Matrix Calculus III Summer 2013, Session II Monday, July 15, 2013 Agenda 1. The rank of a matrix 2. The inverse of a square matrix Gaussian Gaussian solves a linear system by reducing
More informationMATH2210 Notebook 1 Fall Semester 2016/2017. 1 MATH2210 Notebook 1 3. 1.1 Solving Systems of Linear Equations... 3
MATH0 Notebook Fall Semester 06/07 prepared by Professor Jenny Baglivo c Copyright 009 07 by Jenny A. Baglivo. All Rights Reserved. Contents MATH0 Notebook 3. Solving Systems of Linear Equations........................
More information2x + y = 3. Since the second equation is precisely the same as the first equation, it is enough to find x and y satisfying the system
1. Systems of linear equations We are interested in the solutions to systems of linear equations. A linear equation is of the form 3x 5y + 2z + w = 3. The key thing is that we don t multiply the variables
More informationLinear Equations ! 25 30 35$ & " 350 150% & " 11,750 12,750 13,750% MATHEMATICS LEARNING SERVICE Centre for Learning and Professional Development
MathsTrack (NOTE Feb 2013: This is the old version of MathsTrack. New books will be created during 2013 and 2014) Topic 4 Module 9 Introduction Systems of to Matrices Linear Equations Income = Tickets!
More informationLecture Notes 2: Matrices as Systems of Linear Equations
2: Matrices as Systems of Linear Equations 33A Linear Algebra, Puck Rombach Last updated: April 13, 2016 Systems of Linear Equations Systems of linear equations can represent many things You have probably
More informationMethods for Finding Bases
Methods for Finding Bases Bases for the subspaces of a matrix Rowreduction methods can be used to find bases. Let us now look at an example illustrating how to obtain bases for the row space, null space,
More informationNotes on Determinant
ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 918/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without
More informationBasic Terminology for Systems of Equations in a Nutshell. E. L. Lady. 3x 1 7x 2 +4x 3 =0 5x 1 +8x 2 12x 3 =0.
Basic Terminology for Systems of Equations in a Nutshell E L Lady A system of linear equations is something like the following: x 7x +4x =0 5x +8x x = Note that the number of equations is not required
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 informationReduced echelon form: Add the following conditions to conditions 1, 2, and 3 above:
Section 1.2: Row Reduction and Echelon Forms Echelon form (or row echelon form): 1. All nonzero rows are above any rows of all zeros. 2. Each leading entry (i.e. left most nonzero entry) of a row is in
More informationLecture 1: Systems of Linear Equations
MTH Elementary Matrix Algebra Professor Chao Huang Department of Mathematics and Statistics Wright State University Lecture 1 Systems of Linear Equations ² Systems of two linear equations with two variables
More informationby the matrix A results in a vector which is a reflection of the given
Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the yaxis We observe that
More informationMAT 200, Midterm Exam Solution. a. (5 points) Compute the determinant of the matrix A =
MAT 200, Midterm Exam Solution. (0 points total) a. (5 points) Compute the determinant of the matrix 2 2 0 A = 0 3 0 3 0 Answer: det A = 3. The most efficient way is to develop the determinant along the
More informationB such that AB = I and BA = I. (We say B is an inverse of A.) Definition A square matrix A is invertible (or nonsingular) if matrix
Matrix inverses Recall... Definition A square matrix A is invertible (or nonsingular) if matrix B such that AB = and BA =. (We say B is an inverse of A.) Remark Not all square matrices are invertible.
More informationWe seek a factorization of a square matrix A into the product of two matrices which yields an
LU Decompositions We seek a factorization of a square matrix A into the product of two matrices which yields an efficient method for solving the system where A is the coefficient matrix, x is our variable
More information1.5 SOLUTION SETS OF LINEAR SYSTEMS
12 CHAPTER 1 Linear Equations in Linear Algebra 1.5 SOLUTION SETS OF LINEAR SYSTEMS Many of the concepts and computations in linear algebra involve sets of vectors which are visualized geometrically as
More informationDETERMINANTS. b 2. x 2
DETERMINANTS 1 Systems of two equations in two unknowns A system of two equations in two unknowns has the form a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 This can be written more concisely in
More informationSolving Systems of Linear Equations. Substitution
Solving Systems of Linear Equations There are two basic methods we will use to solve systems of linear equations: Substitution Elimination We will describe each for a system of two equations in two unknowns,
More informationLecture 6. Inverse of Matrix
Lecture 6 Inverse of Matrix Recall that any linear system can be written as a matrix equation In one dimension case, ie, A is 1 1, then can be easily solved as A x b Ax b x b A 1 A b A 1 b provided that
More informationSituation 23: Simultaneous Equations Prepared at the University of Georgia EMAT 6500 class Date last revised: July 22 nd, 2013 Nicolina Scarpelli
Situation 23: Simultaneous Equations Prepared at the University of Georgia EMAT 6500 class Date last revised: July 22 nd, 2013 Nicolina Scarpelli Prompt: A mentor teacher and student teacher are discussing
More informationImages and Kernels in Linear Algebra By Kristi Hoshibata Mathematics 232
Images and Kernels in Linear Algebra By Kristi Hoshibata Mathematics 232 In mathematics, there are many different fields of study, including calculus, geometry, algebra and others. Mathematics has been
More informationMATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix.
MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix. Nullspace Let A = (a ij ) be an m n matrix. Definition. The nullspace of the matrix A, denoted N(A), is the set of all ndimensional column
More information2.5 Elementary Row Operations and the Determinant
2.5 Elementary Row Operations and the Determinant Recall: Let A be a 2 2 matrtix : A = a b. The determinant of A, denoted by det(a) c d or A, is the number ad bc. So for example if A = 2 4, det(a) = 2(5)
More informationSYSTEMS OF EQUATIONS AND MATRICES WITH THE TI89. by Joseph Collison
SYSTEMS OF EQUATIONS AND MATRICES WITH THE TI89 by Joseph Collison Copyright 2000 by Joseph Collison All rights reserved Reproduction or translation of any part of this work beyond that permitted by Sections
More informationSection 8.2 Solving a System of Equations Using Matrices (Guassian Elimination)
Section 8. Solving a System of Equations Using Matrices (Guassian Elimination) x + y + z = x y + 4z = x 4y + z = System of Equations x 4 y = 4 z A System in matrix form x A x = b b 4 4 Augmented Matrix
More informationArithmetic and Algebra of Matrices
Arithmetic and Algebra of Matrices Math 572: Algebra for Middle School Teachers The University of Montana 1 The Real Numbers 2 Classroom Connection: Systems of Linear Equations 3 Rational Numbers 4 Irrational
More information5.5. Solving linear systems by the elimination method
55 Solving linear systems by the elimination method Equivalent systems The major technique of solving systems of equations is changing the original problem into another one which is of an easier to solve
More informationLecture Notes: Matrix Inverse. 1 Inverse Definition
Lecture Notes: Matrix Inverse Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong taoyf@cse.cuhk.edu.hk Inverse Definition We use I to represent identity matrices,
More informationMath 215 HW #1 Solutions
Math 25 HW # Solutions. Problem.2.3. Describe the intersection of the three planes u+v+w+z = 6 and u+w+z = 4 and u + w = 2 (all in fourdimensional space). Is it a line or a point or an empty set? What
More informationSolving Systems of Linear Equations
LECTURE 5 Solving Systems of Linear Equations Recall that we introduced the notion of matrices as a way of standardizing the expression of systems of linear equations In today s lecture I shall show how
More information5 Homogeneous systems
5 Homogeneous systems Definition: A homogeneous (homojeen ius) system of linear algebraic equations is one in which all the numbers on the right hand side are equal to : a x +... + a n x n =.. a m
More information4.5 Linear Dependence and Linear Independence
4.5 Linear Dependence and Linear Independence 267 32. {v 1, v 2 }, where v 1, v 2 are collinear vectors in R 3. 33. Prove that if S and S are subsets of a vector space V such that S is a subset of S, then
More informationSolution. Area(OABC) = Area(OAB) + Area(OBC) = 1 2 det( [ 5 2 1 2. Question 2. Let A = (a) Calculate the nullspace of the matrix A.
Solutions to Math 30 Takehome prelim Question. Find the area of the quadrilateral OABC on the figure below, coordinates given in brackets. [See pp. 60 63 of the book.] y C(, 4) B(, ) A(5, ) O x Area(OABC)
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 informationLecture Note on Linear Algebra 15. Dimension and Rank
Lecture Note on Linear Algebra 15. Dimension and Rank WeiShi Zheng, wszheng@ieee.org, 211 November 1, 211 1 What Do You Learn from This Note We still observe the unit vectors we have introduced in Chapter
More informationSolution of Linear Systems
Chapter 3 Solution of Linear Systems In this chapter we study algorithms for possibly the most commonly occurring problem in scientific computing, the solution of linear systems of equations. We start
More information1 Eigenvalues and Eigenvectors
Math 20 Chapter 5 Eigenvalues and Eigenvectors Eigenvalues and Eigenvectors. Definition: A scalar λ is called an eigenvalue of the n n matrix A is there is a nontrivial solution x of Ax = λx. Such an x
More informationUsing row reduction to calculate the inverse and the determinant of a square matrix
Using row reduction to calculate the inverse and the determinant of a square matrix Notes for MATH 0290 Honors by Prof. Anna Vainchtein 1 Inverse of a square matrix An n n square matrix A is called invertible
More information4.34.4 Systems of Equations
4.34.4 Systems of Equations A linear equation in 2 variables is an equation of the form ax + by = c. A linear equation in 3 variables is an equation of the form ax + by + cz = d. To solve a system of
More informationQuestion 2: How do you solve a matrix equation using the matrix inverse?
Question : How do you solve a matrix equation using the matrix inverse? In the previous question, we wrote systems of equations as a matrix equation AX B. In this format, the matrix A contains the coefficients
More information1 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 informationAbstract: We describe the beautiful LU factorization of a square matrix (or how to write Gaussian elimination in terms of matrix multiplication).
MAT 2 (Badger, Spring 202) LU Factorization Selected Notes September 2, 202 Abstract: We describe the beautiful LU factorization of a square matrix (or how to write Gaussian elimination in terms of matrix
More informationSystems of Linear Equations
Chapter 1 Systems of Linear Equations 1.1 Intro. to systems of linear equations Homework: [Textbook, Ex. 13, 15, 41, 47, 49, 51, 65, 73; page 11]. Main points in this section: 1. Definition of Linear
More informationSec 4.1 Vector Spaces and Subspaces
Sec 4. Vector Spaces and Subspaces Motivation Let S be the set of all solutions to the differential equation y + y =. Let T be the set of all 2 3 matrices with real entries. These two sets share many common
More informationLecture 3: Finding integer solutions to systems of linear equations
Lecture 3: Finding integer solutions to systems of linear equations Algorithmic Number Theory (Fall 2014) Rutgers University Swastik Kopparty Scribe: Abhishek Bhrushundi 1 Overview The goal of this lecture
More informationMatrices, Determinants and Linear Systems
September 21, 2014 Matrices A matrix A m n is an array of numbers in rows and columns a 11 a 12 a 1n r 1 a 21 a 22 a 2n r 2....... a m1 a m2 a mn r m c 1 c 2 c n We say that the dimension of A is m n (we
More information1 Solving LPs: The Simplex Algorithm of George Dantzig
Solving LPs: The Simplex Algorithm of George Dantzig. Simplex Pivoting: Dictionary Format We illustrate a general solution procedure, called the simplex algorithm, by implementing it on a very simple example.
More informationLS.6 Solution Matrices
LS.6 Solution Matrices In the literature, solutions to linear systems often are expressed using square matrices rather than vectors. You need to get used to the terminology. As before, we state the definitions
More informationName: Section Registered In:
Name: Section Registered In: Math 125 Exam 3 Version 1 April 24, 2006 60 total points possible 1. (5pts) Use Cramer s Rule to solve 3x + 4y = 30 x 2y = 8. Be sure to show enough detail that shows you are
More informationINTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL
SOLUTIONS OF THEORETICAL EXERCISES selected from INTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL Eighth Edition, Prentice Hall, 2005. Dr. Grigore CĂLUGĂREANU Department of Mathematics
More informationSYSTEMS OF EQUATIONS
SYSTEMS OF EQUATIONS 1. Examples of systems of equations Here are some examples of systems of equations. Each system has a number of equations and a number (not necessarily the same) of variables for which
More informationSECTION 81 Systems of Linear Equations and Augmented Matrices
86 8 Systems of Equations and Inequalities In this chapter we move from the standard methods of solving two linear equations with two variables to a method that can be used to solve linear systems with
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 informationMATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.
MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V V (x,y) x + y V and scalar
More informationMATH10212 Linear Algebra B Homework 7
MATH22 Linear Algebra B Homework 7 Students are strongly advised to acquire a copy of the Textbook: D C Lay, Linear Algebra and its Applications Pearson, 26 (or other editions) Normally, homework assignments
More informationUniversity of Lille I PC first year list of exercises n 7. Review
University of Lille I PC first year list of exercises n 7 Review Exercise Solve the following systems in 4 different ways (by substitution, by the Gauss method, by inverting the matrix of coefficients
More informationMath 313 Lecture #10 2.2: The Inverse of a Matrix
Math 1 Lecture #10 2.2: The Inverse of a Matrix Matrix algebra provides tools for creating many useful formulas just like real number algebra does. For example, a real number a is invertible if there is
More informationLinear Equations in Linear Algebra
1 Linear Equations in Linear Algebra 1.5 SOLUTION SETS OF LINEAR SYSTEMS HOMOGENEOUS LINEAR SYSTEMS A system of linear equations is said to be homogeneous if it can be written in the form A 0, where A
More informationThe Inverse of a Square Matrix
These notes closely follow the presentation of the material given in David C Lay s textbook Linear Algebra and its Applications (3rd edition) These notes are intended primarily for inclass presentation
More informationSolutions to Review Problems
Chapter 1 Solutions to Review Problems Chapter 1 Exercise 42 Which of the following equations are not linear and why: (a x 2 1 + 3x 2 2x 3 = 5. (b x 1 + x 1 x 2 + 2x 3 = 1. (c x 1 + 2 x 2 + x 3 = 5. (a
More informationLinear Codes. In the V[n,q] setting, the terms word and vector are interchangeable.
Linear Codes Linear Codes In the V[n,q] setting, an important class of codes are the linear codes, these codes are the ones whose code words form a subvector space of V[n,q]. If the subspace of V[n,q]
More informationNOTES ON LINEAR TRANSFORMATIONS
NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
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 information8 Square matrices continued: Determinants
8 Square matrices continued: Determinants 8. Introduction Determinants give us important information about square matrices, and, as we ll soon see, are essential for the computation of eigenvalues. You
More informationAu = = = 3u. Aw = = = 2w. so the action of A on u and w is very easy to picture: it simply amounts to a stretching by 3 and 2, respectively.
Chapter 7 Eigenvalues and Eigenvectors In this last chapter of our exploration of Linear Algebra we will revisit eigenvalues and eigenvectors of matrices, concepts that were already introduced in Geometry
More informationII. Linear Systems of Equations
II. Linear Systems of Equations II. The Definition We are shortly going to develop a systematic procedure which is guaranteed to find every solution to every system of linear equations. The fact that such
More information4. MATRICES Matrices
4. MATRICES 170 4. Matrices 4.1. Definitions. Definition 4.1.1. A matrix is a rectangular array of numbers. A matrix with m rows and n columns is said to have dimension m n and may be represented as follows:
More informationSolving Linear Diophantine Matrix Equations Using the Smith Normal Form (More or Less)
Solving Linear Diophantine Matrix Equations Using the Smith Normal Form (More or Less) Raymond N. Greenwell 1 and Stanley Kertzner 2 1 Department of Mathematics, Hofstra University, Hempstead, NY 11549
More informationHomogeneous systems of algebraic equations. A homogeneous (homogeen ius) system of linear algebraic equations is one in which
Homogeneous systems of algebraic equations A homogeneous (homogeen ius) system of linear algebraic equations is one in which all the numbers on the right hand side are equal to : a x + + a n x n = a
More informationVector Spaces 4.4 Spanning and Independence
Vector Spaces 4.4 and Independence October 18 Goals Discuss two important basic concepts: Define linear combination of vectors. Define Span(S) of a set S of vectors. Define linear Independence of a set
More information160 CHAPTER 4. VECTOR SPACES
160 CHAPTER 4. VECTOR SPACES 4. Rank and Nullity In this section, we look at relationships between the row space, column space, null space of a matrix and its transpose. We will derive fundamental results
More informationx y The matrix form, the vector form, and the augmented matrix form, respectively, for the system of equations are
Solving Sstems of Linear Equations in Matri Form with rref Learning Goals Determine the solution of a sstem of equations from the augmented matri Determine the reduced row echelon form of the augmented
More informationMA 242 LINEAR ALGEBRA C1, Solutions to Second Midterm Exam
MA 4 LINEAR ALGEBRA C, Solutions to Second Midterm Exam Prof. Nikola Popovic, November 9, 6, 9:3am  :5am Problem (5 points). Let the matrix A be given by 5 6 5 4 5 (a) Find the inverse A of A, if it exists.
More information4.6 Null Space, Column Space, Row Space
NULL SPACE, COLUMN SPACE, ROW SPACE Null Space, Column Space, Row Space In applications of linear algebra, subspaces of R n typically arise in one of two situations: ) as the set of solutions of a linear
More information12 Greatest Common Divisors. The Euclidean Algorithm
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 12 Greatest Common Divisors. The Euclidean Algorithm As mentioned at the end of the previous section, we would like to
More informationLinear Algebra Notes
Linear Algebra Notes Chapter 19 KERNEL AND IMAGE OF A MATRIX Take an n m matrix a 11 a 12 a 1m a 21 a 22 a 2m a n1 a n2 a nm and think of it as a function A : R m R n The kernel of A is defined as Note
More informationa = bq + r where 0 r < b.
Lecture 5: Euclid s algorithm Introduction The fundamental arithmetic operations are addition, subtraction, multiplication and division. But there is a fifth operation which I would argue is just as fundamental
More information2.1 Functions. 2.1 J.A.Beachy 1. from A Study Guide for Beginner s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair
2.1 J.A.Beachy 1 2.1 Functions from A Study Guide for Beginner s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair 21. The Vertical Line Test from calculus says that a curve in the xyplane
More informationElementary Number Theory We begin with a bit of elementary number theory, which is concerned
CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,
More informationMatrix Algebra and Applications
Matrix Algebra and Applications Dudley Cooke Trinity College Dublin Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 1 / 49 EC2040 Topic 2  Matrices and Matrix Algebra Reading 1 Chapters
More informationLINEAR ALGEBRA. September 23, 2010
LINEAR ALGEBRA September 3, 00 Contents 0. LUdecomposition.................................... 0. Inverses and Transposes................................. 0.3 Column Spaces and NullSpaces.............................
More information2.1: MATRIX OPERATIONS
.: MATRIX OPERATIONS What are diagonal entries and the main diagonal of a matrix? What is a diagonal matrix? When are matrices equal? Scalar Multiplication 45 Matrix Addition Theorem (pg 0) Let A, B, and
More information