Homework: 2.1 (page 56): 7, 9, 13, 15, 17, 25, 27, 35, 37, 41, 46, 49, 67


 Patience Terry
 1 years ago
 Views:
Transcription
1 Chapter Matrices Operations with Matrices Homework: (page 56):, 9, 3, 5,, 5,, 35, 3, 4, 46, 49, 6 Main points in this section: We define a few concept regarding matrices This would include addition of matrices, scalar multiplication and multiplication of matrices We also represent a system of linear equation as equation with matrices 4
2 4 CHAPTER MATRICES In this section, we will do some algebra of matrices That means, we will add, subtract, multiply matrices Matrices will usually be denoted by upper case letters, A, B, C, Such a matrix A is also denoted by a ij a a a 3 a n a a a 3 a n a 3 a 3 a 33 a 3n a m a m a m3 a mn Remark I used parentheses in the last chapter, not a square brackets The textbook uses square brackets Definition Two matrices A a ij and B b ij are equal if both A and B have same size m n and the entries a ij b ij for all i m and j n Read Textbook, Example, p 4 for examples of unequal matrices Definition Following are some standard terminologies: A matrix with only one column is called a column matrix or column vector For example, 3 a 9 is a column matrix A matrix with only one row is called a row matrix or row vector For example b is a row matrix 3
3 OPERATIONS WITH MATRICES 43 3 Bold face lower case letters, as above, will often be used to denote row or column matrices Matrix Addition Definition 3 We difine addition of two matrices of same size Suppose A a ij and B b ij be two matrices of same size m n Then the sum A + B is defined to be the matrix of size m n given by A + B a ij + b ij For example, with A , B we have A+B Also, for example, with 35 C, D we have C+D While, the sum A + C is not defined because A and C do not have same size 4 Read Textbook, Example, p 48 for more such examples Scalar Multiplication Recall, in some contexts, real numbers are referred to as scalars (in contrast with vectors) We define, multiplication of a matrix A by a scalar c
4 44 CHAPTER MATRICES Definition 4 Let A a ij be an m n matrix and c be a scalar We define Scalar multiple ca of A by c as the matrix of same size given by ca ca ij For a matrix A, the negative of A denotes ( )A Also A B : A + ( )B Let c and A Then, with c we have ca , B Likewise, A B A + ( )B ( ) Read Textbook, Example 3, p 49 for more such computations
5 OPERATIONS WITH MATRICES 45 3 Matrix Multiplication The textbook gives a helpful motivation for defining matrix multiplication in page 49 Read it, if you are not already motivated Definition 5 Suppose A a ij is a matrix of size m n and B b ij is a matrix of size n p Then the product AB c ij is a matrix size m p where c ij : a i b j + a i b j + a i3 b 3j + + a in b nj n a ik b kj k Note that number of columns (n) of A must be equal to number of rows (n) of B, for the product AB to be defined Note the number of rows of AB is equal to is same as that (m) of A and number of columns of AB is equal to is same as that (n) of B 3 Read Textbook, Example 45, p55 I will workout a few below Exercise 6 (Ex, p 5) Let A 8 3 Compute AB abd BA We have AB, B 3 + ( ) + + ( ) + ( 3) + ( ) + + ( ) ( ) + 8 ( 3) + ( ) ( ) ( ) ( 3) ( )
6 46 CHAPTER MATRICES Now, we compute BA We have ( ) + ( ) ( BA ( ) + ( ) ( + ( 3) + 3 ( ) + ( 3) ( ) + + ( 3) 8 + ( Also note that AB BA Exercise (Ex 6, p 5) Let 0 3 A Compute AB and BA, if definded, and B Solution: Note BA is not defined, because size of B is 3 and size of A is 3 3 But AB is defined and is a 3 matrix (or a column matrix) We have 0 + ( ) ( 3) + 3 AB ( 3) ( ) ( 3) Exercise 8 (Ex 8 p 5) Let A and B 6 4
7 OPERATIONS WITH MATRICES 4 Compute AB and BA, if definded Solution Note AB is not defined, because size of A is 5 and size of B is But BA is defined and we have BA ( ) + 6 ( ) ( ) + ( ) Systems of linnear equations Systems of linear equations can be represented in a matrix form Given a system of liner equations a x + a x + a 3 x a n x n b a x + a x + a 3 x a n x n b a 3 x + a 3 x + a 33 x a 3n x n b 3 a m x + a m x + a m3 x a mn x n b m we can write this system in the matrix form as where A a a a 3 a n a a a 3 a n a 3 a 3 a 33 a 3n a m a m a m3 a mn Ax b, x x x x 3 x n, b Clearly, A is the coefficient matrix, x would be called the unknown (or variable) matrix (or vector) and b would be called the constant vector (or matrix) b b b 3 b n
8 48 CHAPTER MATRICES With a a a a 3 a m, a we can write (or think) as a matrix of matrices a a a 3 a m, a j a j a j a 3j a mj A a a a j a n, a n The above system of linear equations can be (re)writeen as x a + x a + + x j a j + + x n a n b a n a n a 3n a mn We say, b is a linear combination of the column metrices (or vectors) a, a,, a n with coefficents x, x,, x n 3 So, the solutions of the above system are precisely those c,, c n such that b is a linear combination of the vectors a, a,, a n with coefficents c, c,, c n Exercise 9 (Ex 34 (changed), p 58) Consider the system of equation: x +x 3x 3 x +x x x +x 3 Write this system in the matrixform Ax b and solve matrix equation for x Solution: The equation reduces to 3 0 x x x 3
9 OPERATIONS WITH MATRICES 49 This answers the first part system is To solve, the augmented matrix of the 3 0 Using TI84, we can reduce it to GaussJordan form: The corresponding linear system is: Multiplying, this give the solution: x x x 3 x x x Exercise 0 (Ex 40, 58) a c b d Solution: Here we have four unknowns a, b, c, d In any case, multiplying: a + 3b a + b 3 c + 3d c + d 4
10 50 CHAPTER MATRICES Equating respective entries: a +3b 3 a +b c +3d 4 c +d The augmented matrix of the system: Use TI84 to reduce it to the GaussJordan form: Looking at the corresponding to this matrix, we get a b a 48, b 3, c, d 6 and c d
11 PROPERTIES OF MATRIX OPERATIONS 5 Properties of Matrix operations Homework: (page 0):, 3,, 9, 5,,, 3, 9, 3, 35, 3, 39, 43, 5, 59 Main points in this section: We write down some of the properties of matrix addition and multiplication We define transpose of a matrix 3 We start writing some proofs
12 5 CHAPTER MATRICES We start developing some algebra of matrices Theorem (Properties) Let A, B, C be three m n matrices and c, d be scalars Then A + B B + A Commutativity of matrix addition (A + B) + C A + (B + C) Associativity of matrix addition 3 (cd)a c(da) Associativity of scalar multiplication 4 A A identity of scalar multiplication 5 c(a + B) ca + cb Distributivity of scalar multiplication 6 (c + d)a ca + da Distributivity of scalar multiplication Proof One needs to prove all these statements using definitions of addition and scalar multiplication To prove the commutativity of matrix addition (), Let A a ij, B b ij Both A and B have same size m n, so A + B, B + A are defined From definition A + B a ij + b ij a ij + b ij and B + A b ij + a ij b ij + a ij From commutative property of addition of real numbers, we have a ij + b ij b ij +a ij Therefore, from definition of equality of matrices, A+B B + A So, () is proved Other properties are proved similarly Remark For matrices A, B, C as in the theorem, by the expression A + B + C we mean (A + B) + C or A + (B + C) It is well defined, because (A + B) + C A + (B + C) by associatieve property() of matrix addition Theorem Let O mn denote the m n matrix whose entries are all zero Let A be a m n matrix Then Then A + O mn A
13 PROPERTIES OF MATRIX OPERATIONS 53 We have A + ( A) O mn 3 If ca O mn, then either c 0 or A 0 We say, on the set of all m n matrices, O mn is an additive identity (property ()), and ( A) is the additive inverse of A (property ()) Properties of matrix multiplication Theorem 3 (MultProperties) Let A, B, C be three matrices (of varying sizes) so that all the products below are defined and c be a scalars Then AB BA Failure of Commutativity of multiplication (AB)C A(BC) Associativity of multiplication 3 (A + B)C AC + BC Left Distributivity of multiplication 4 A(B + C) AB + AC Right Distributivity of multiplication 5 c(ab) (ca)b an Associativity In (), by AB BA, we mean AB is not always equal to BA Proof We will only prove (4) In this case, let A be a matrix of size m n and then B, C would have to be of same size n p Write A a a a 3 a n a a a 3 a n a 3 a 3 a 33 a 3n a m a m a m3 a mn,
14 54 CHAPTER MATRICES and b b b 3 b p c c c 3 c p b b b 3 b p c c c 3 c p B b 3 b 3 b 33 b 3p, C c 3 c 3 c 33 c 3p b n b n b n3 b np c n c n c n3 c np Therefore, A(B + C) a a a 3 a n a a a 3 a n a 3 a 3 a 33 a 3n b + c b + c b 3 + c 3 b p + c p b + c b + c b 3 + c 3 b p + c p b 3 + c 3 b 3 + c 3 b 33 + c 33 b 3p + c 3p a m a m a m3 a mn b n + c n b n + c n b n3 + c n3 b np + c np which is n k a k(b k + c k ) n k a k(b k + c k ) n k a k(b kp + c kp ) n k a k(b k + c k ) n k a k(b k + c k ) n k a k(b kp + c kp ) n k a 3k(b k + c k ) n k a 3k(b k + c k ) n k a 3k(b kp + c kp ) n k a mk(b k + c k ) n k a mk(b k + c k ) n k a mk(b kp + c kp )
15 PROPERTIES OF MATRIX OPERATIONS 55 which is n k a n kb k k a kb k n k a kb kp n k a n kb k k a kb k n k a kb kp n k a n 3kb k k a 3kb k n k a 3kb kp n k a n mkb k k a mkb k n k a mkb kp + n k a kc k n k a kc k n k a kc kp AB + AC n k a n kc k k a kc k n k a kc kp n k a n 3kc k k a 3kc k n k a 3kc kp n k a mkc k ) n k a mkc k n k a mkc kp So, A(B + C) AB + AC and the proof of complete Alternate way to write the same proof: Let A be a matrix of size m n and then B, C would have to be of same size n p Write A a ik, B b kj, C c kj Then B + C b kj + c kj So, A(B + C) a ik b kj + c kj α ij (say) Then the (ij) th entry α ij of A(B + C) is given by n n n α ij a ik (b kj + c kj ) a ik b kj + a ik c kj k k k But n n AB a ik b kj, and AC a ik c kj k k So, the (ij) th entry of AB + AC is also equal to α ij Therefore A(B + C) AB + AC and the proof is complete
16 56 CHAPTER MATRICES Remark For matrices A, B, C as in the theorem so that all the multiplications in the associative law () are defined Then the expression ABC would mean (AB)C or A(BC) It is well defined, because (AB)C A(BC) by associatieve property() of matrix multiplication Reading assignment Textbook, Example 3, p 64 to experience that associativity (property ) works Textbook, Example 4, p 64 to see examples that commutativity for multiplication fails (property ) 3 Textbook, Example 5, p 65 to see examples that cancellation fails That means, examples AB AC for nonzero A, B, C with B C This makes solving matrix equation like AX AC, different from solving equation in real numbers, like ax c Theorem 4 Let I n denote the square matrix of order n whose main diagonal entries are and all the entries are zero So, I n 0 0 0, I 3 0 0, I Let A be an m n matrix Then I m A A and AI n A Because of these two properties, I n is called the identity matrix of order n Proof Easy
17 PROPERTIES OF MATRIX OPERATIONS 5 Reading assignment Textbook, Example 6, p 66 Textbook, Example, p 66, just to get used to computing A r Following the definition in item (4) we gave a classification of system of equation, which we state as a theorem and prove as follows Theorem 5 Let Ax b be a linear system of m equations in n unknowns Then exactly one of the follwoing is true: The system has no solution The system has exactly one solution 3 The system has infinitely many solutions Proof If the system has no solution or have exactly one solution, then there is nothing to prove So, assume, it has at least two distinct solutions x, x So, Ax b and Ax b Subtracting the second from the first, we have A(x x ) 0 Write y x x Then y 0 and Ay 0 So, for any scalar c, we have A(x + cy) Ax + A(cy) b + (Ac)y b + c(ay) b + c0 b So, x +cy is a solution, for each scalar c (and they are all distinct) So, the system has infinitely many solutuions This completes the proof Transpose of a matrix Definition 6 The transpose of a matrix A is obtained by writing the rows as columns (and/or writing the columns as rows) The transpose of A is denoted bt A T So, for the matrix
18 58 CHAPTER MATRICES A a a a 3 a n a a a 3 a n a 3 a 3 a 33 a 3n A T a a a 3 a m a a a 3 a m a 3 a 3 a 33 a m3 a m a m a m3 a mn a n a n a 3n a mn Definition A (square) matrix A is said to be symmetric, if A A T Theorem 8 Let A, B be matrices (of varying sizes so that the sum or product is defined) and c be a scalar Then, (A T ) T A (That means transpose of transpose is the same) (A + B) T A T + B T 3 (ca) T ca T 4 (AB) T B T A T (This is the most important one in this list) In fact, properties, 4 extends to sum of higher number of matrices Reading assignment Textbook, Example 8, p 68 to compute transpose of a matrix Textbook, Example 9, p 66, about transpose and product of matrices
19 PROPERTIES OF MATRIX OPERATIONS 59 Exercise 9 (Ex 8, p 0) Let A 0 Then Solve X 3A B 3 4, B Solution: In this case X 3A B 3 0 which is Solve X A B Solution: We have X A B 0 OR 3 4 X Divide by (ie multiply by ), we have X
20 60 CHAPTER MATRICES 3 Solve X + 3A B Solution: We have X + 3A B Subtracting 3A from both sides, X B 3A So, 0 3 X B 3A Divide by (ie multiply by ), we have X Exercise 0 (Ex 0changed, p 0) Let 3 3 A, B, C 0 Compute C(BA) and A(BC), if defined 0 0 Solution: Note A(BC) is not defined because A has 3 column and BC has two rows We compute C(BA) We have C(BA) (CB)A and 0 3 CB 0 3 So, C(BA) (CB)A Exercise (Ex 8, p 0) Let 4 A, B 4 05
21 PROPERTIES OF MATRIX OPERATIONS 6 Then AB 0 but A 0 nor B 0 (Note, such a phenomenon will not occur with real numbers That is why with real numbers, ax 0, a 0 x 0 We cannot do similar algebra with matrices With A 0 the solution to the equation AX 0 is not necessarily X 0) Solution: Obvious Exercise (Ex, p 0) Let A, and I I 0 Compute A + IA 0 0 Solution: We have A + IA A + A A Exercise 3 (Ex 4 p 0) Let A 3 4 Compute A T, A T A and AA T 0 Solution: We have A T So, A T A
22 6 CHAPTER MATRICES Also AA T which is Exercise and comment: Notice in this exercise 4 above, both AA T and A T A are symmetric matrices This is not a surprise In fact, for any matrix A, both AA T and A T A are symmetric matrices Proof ( AA T ) T ( A T ) T A T AA T So, AA T is symmetric Similarly, A T A is symmetric Exercise 4 (Ex 38 p ) Let X, Y 0, Z 4 3 4, W Find scalars a, b such that Z ax + by 0 0, O Solution: This is, in fact, a problem of solving a system of linear equations Suppose Z ax + by Then, a a X Y Z OR 0 4 b b 3 4
23 PROPERTIES OF MATRIX OPERATIONS 63 Here a, b are unknown to be solved for The augmented matrix of this system is: By TI84, the matrix reduced to the GaussJordan form: This gives a, b (We can check X Y Z) Show that there do not exist scalar a, b such that W ax + by Solution: This is, in fact, a problem of solving a system of linear equations Suppose W ax + by Then, 0 a a X Y W OR 0 0 b b 3 Here a, b are unknown to be solved for The augmented matrix of this system is: Here a, b are unknown to be solved for The 3 augmented matrix of this system is: By TI84, the matrix reduced to the GaussJordan form:
24 64 CHAPTER MATRICES The system corresponding to this matrix is a 0, b 0, 0, which is inconsistent 3 Show that ax + by + cw O then a b c 0 Solution: This is, again, a problem of solving a system of linear equations Suppose O ax + by + cw Then, X Y W a b c O OR 0 a 0 0 b 3 The augmented matrix of this system is given by c From TI83/84, the matrix reduces to the following GaussJordan form: So, a 0, b 0, c 0, as was required to show 4 Find the scalars a, b, c, not all zero (nontrivial solution), such that ax + by + cz O Solution: This is, again, a problem of solving a system of linear equations Suppose O ax + by + cz Then, X Y Z a b c O OR a 0 4 b 3 4 c
25 PROPERTIES OF MATRIX OPERATIONS 65 The augmented matrix of this system is given by From TI83/84, the matrix reduces to the following GaussJordan form: The linear system corresponding to this matrix is a +c 0 b c Using c t as parameter, we have a t, b t, c t So, for any scalar t, we have tx + ty + tz O Exercise 5 (Ex 44, p ) Let f(x) x x + 6 and A Compute f(a) 5 4 Solution: We have A AA So, f(a) A A + 6I
26 66 CHAPTER MATRICES So, f(a) O (One can say that A is a (matrix) root of f(x))
27 3 THE INVERSE OF A MATRIX 6 3 The Inverse of a matrix Homework: Textbook, Ex 9, 3, 5,, 33, 35, 3, 4, 45 ; page 84 Main point in the section is to define and compute inverse of matrices: We give a formula to compute inverse of a matrices We describe the method of row reduction to compute inverse of a matrix of any size 3 We use inverse of matrices to solve system of linear equations
28 68 CHAPTER MATRICES Definition 3 For a square matrix A of size n n, we say that A is invertible (or nonsingular) if there is an n n, matrix B such that AB BA I n where I n is the identity matrix of order n The matrix B is called the (multiplicative ) inverse of A If a matrix does not have an inverse, it is called a noninvertible (or singular) matrix A nonsquare matrix does have an inverse Because if A is an m n matrix with m n, then for AB and BA to be defined, the size of B has to be n m The size of AB would be m m and size of BA would be n n So, AB BA Theorem 3 If A is invertible, then the inverse is unique inverse is denoted by A The Proof Suppose B and C are two inverses of A We nned to prove B C We have So, So, the proof is complete AB BA I and AC CA I B BI B(AC) (BA)C IC C Remark () Note that the proof works if C was only a rightinverse and B was a leftinverse of A () Also note that the proof did not use any property of matrices, except associativity 3 Finding Inverse of a matrix There are many methods of finding inverse of a matrix We will discuss at least two of them The easy one, for a matrices is given by the theorem:
29 3 THE INVERSE OF A MATRIX 69 Theorem 33 Suppose A a c b d is a nonzero matrix Then If ad bc 0, then A has no invierse (ie A is a singular matrix) If ad bc 0, then A ad bc d c b a Proof Write B d c b a First, a b AB c d d c b a ad bc 0 0 ad bc (ad bc)i (Case :) Assume ad bc 0 Then, we have AB O So, A cannot have an inverse (otherwise we will get B A (AB) O, which is not the case) (Case :) Assume ad bc 0 We need to prove A( B) ad bc I ( B)A Multiply the above equation by, we get ad bc ad bc ( ) A ad bc B I Similarly, ( ad bc B) A I So, the proof is complete Exercise 34 (Ex 6, p84) Find the inverse of the matrix A, if exists 3
30 0 CHAPTER MATRICES Solution: We have ad bc ( 3) ( ) So, by theorem 33, A ad bc d c b a 3 Reading assignment: Read Textbook, Example and, p 45 and Textbook, Example 5, p 9 3 nd method of finding inverse of a matrix The second mothod of finding the inverse evolves out of the spirit of solving equation Suppose A is an n n matrix and it has an inverse X, where X is an n n matrix To find X, we have to solve the equation AX I n The following method is suggested to solve this equation: Write the augmented matrix of this Equation AX I n as A I n If possible, row reduce A to the identity matrix I using elementary row operations on the entire augmented matrix A I If the result is I C, then A C If this is not possible, the matrix is not invertible (or singular) 3 Check your work by multiplying AA and A A to see AA A A I Justifcation or Proof Suppose B is an m n matrix Then, each elemetary row operation on a matrix B is same as leftmultiplication of B by a matrix: For example, multiplying the i th row B by a constant is same as leftmultiplication of B by the diagonal matrix of order n, whose (i, i) entry is c and other digonal entries are (and rest of the entries are zero)
31 3 THE INVERSE OF A MATRIX Interchanging of (for example) first and second row is leftmultiplication of B by the (partitioned) matrix: 0 S O 0 O I m We use the notation O to denote the zero matrix of appropriate size So, we mean SB is the matrix obtained by switching first and second row of B 3 Adding c times first row to the second is same as leftmultiplication of B by the (partitioned) matrix: 0 E O c O I m So, we mean EB is the matrix obtained from B by adding c times the first row to second We can write down a similar fact for adding c times the i th row to j th row Our method says, thee is sequence of matrices E, E,, E r such that the leftmuitiplication(s) gives With E E E E r, we have (E E E r )A I I C EA I I C EA E I C This means, EA I and E C So, E C is the (left) inverse of A This completes the justification/proof Reading assignment: Read Textbook, Example 3 and 4, p 65 Exercise 35 (Ex 0, p 84) Compute the inverse of the matrix 3 A 3 9 4
32 CHAPTER MATRICES Solution: Augment this matrix with the idetity matrix I 3 and we have According to the above method, we will try to reduce first 3 3 part to I 3, using row operations Subtract 3 times first row from second and add first roe to third: Now subtract times second row from first and add times second row to third: Divide third row by 4, we get Subtract 3 times third row from first: So, A
33 3 THE INVERSE OF A MATRIX 3 Exercise 36 (Ex, p 84) Compute the inverse of the matrix 0 5 A Solution: Augment this matrix with the idetity matrix I 3 and we have According to the above method, we will try to reduce the left 3 3 part to I 3, using row operations Divide first row by 0: Add 5 times first row to second and substract 3 times first row from third: Divide second row by 35: Subtract 5 times second row from both the first and third:
34 4 CHAPTER MATRICES Multiply last row by 35: Subtract 54 times third row from second and add 0 from the first: times third row So, A Properties of Inverses Following is a list properties of inverses: Theorem 3 Suppose A is an invertible matrix and c 0 is a scalar Also, let k be a positive integer Then (A ) A (A k ) (A ) k 3 (ca) c A 4 (A T ) (A ) T
35 3 THE INVERSE OF A MATRIX 5 Proof From definitions, B is an inverse of A, if AB BA I So, () is obvious To prove (), we need to show (A k )(A ) k ) (A ) k )A k I For k, we need to prove (A )(A ) ) (A ) )A I But (A )(A ) ) (AA)(A A ) A((AA )A ) A(IA ) AA I Similarly, (A ) )A I For any integer k >, we prove it similarly or we can prove it by method of induction: To do this we assume that (A ) k is the inverse of A k and use it to prove that(a ) k+ is the inverse of A k+, as follows: (A k+ )(A ) k+ A k (AA )(A ) k A k I(A ) k A k (A ) k I So, () is proved Also ( ) (ca) c A (c c )(AA ) I I Similarly, ( c A ) (ca) I So, (3) is proved Finally, A T ( A ) T ( A A ) T I T I Similarly, (A ) T A T I So, all the proofs are complete Reading Assignment: Read Textbook, Example 6, p 808 Theorem 38 Let A, B be two invertible matrices of order n Then AB is invertible and (AB) B A Proof We need to prove, (AB) ( B A ) I ( B A ) (AB) But (AB) ( B A ) A (( BB ) A ) A ( IA ) AA I Similarly, we prove (B A ) (AB) I The proof is complete
36 6 CHAPTER MATRICES Theorem 39 (Cancellation Property) Let A, B be two invertible matrices of order n and C be an invertible matrix of the same order Then AC BC A B and CA CB A B Proof Suppose AC BC Multiply this equation by C from the right side (we can do this because it is given that C has an inverse), we get (AC)C (BC)C So A(CC ) B(CC ) So AI BI So A B Similaraly, we prove the other one The proof is complete Theorem 30 Let A be an invertible matrices of order n Then the system of linear equations Ax b has a unique solution given by x A b Proof (Proof is exactly same as that of theorem 39) Suppose Ax b Multiply this eqyuation by A, and we get (A A)x A b Therefore Ix A b Hence x A b The proof is complete Exercise 3 (Ex 6, p 85) Solve the following three linear systems Solve the linear system of equations: x y 3 x +y Solution: With A x x y b 3
37 3 THE INVERSE OF A MATRIX the system can be written as Ax b By theorem 33, A 4 The solution is x A b Solve the linear system of equations: x y x +y 3 Solution: The system can be writeen as Ax b where A, x are same as in () and b W e already computed A 3 4 The solution is x A b Solve the linear system of equations: x y 6 x +y 0 Solution: The system can be writeen as Ax b where A, x are same as in () and 6 b W e already computed A 0 4
38 8 CHAPTER MATRICES The solution is x A b Exercise 3 (Ex 8, p 85) Solve the following two linear systems Solve the linear system of equations: Solution: With 3 A x +x 3x 3 0 x x +x 3 0 x x x 3 x x x x 3 b the system can be written as Ax b We need to find the inverse of A To do this augment the identity matrix I 3, to A and we get Subtract first row from second and third: Divide second row by 3, we get
39 3 THE INVERSE OF A MATRIX 9 Subtract second row from first and add times second to third row: Multiply third row by 3, we get Add 5 3 times third row to first and add 4 3 second: times third row to By theorem 33, A The solution is x A b Solve the linear system of equations: x +x 3x 3 x x +x 3 x x x 3 0
40 80 CHAPTER MATRICES Solution: With A and x as in () and with b the system can be written as Ax b We already computed A So, the solution is x A b Exercise 33 (Ex 34, p 85) Suppose A, B arer two matrices and A 3 Compute (AB) and B Solution: By theorem 38, we have (AB) B A Compute (A T ) Solution: By theorem 3, we have (A T ) (A ) T ( 3 ) T 3
41 4 ELEMENTARY MATRICES 8 3 Compute A (In page 80, for a positive integer k it was defined A k : (A ) k ) We have A A A Compute (A) Solution: By theorem 3, we have (A) A Exercise 34 (Ex 38, p 85) Let x A Find x, so that A is its own inverse Solution: If A it its own inverse then A I So, we have x x 0 0 Multiplying, we have 4 x x 0 0 so 4 x ; so x 3 4 Elementary Matrices We skip this section I included some of it in subsection 3
42 8 CHAPTER MATRICES 5 Application of matrix operations Homework: Textbook, 5 Ex 5,, 9,, 3; p 3 Main point in this section is to do some applications of matrices We discuss stochastic matrices But we will not go deeper into it We discuss application of matrices in Cryptography
43 5 APPLICATION OF MATRIX OPERATIONS 83 5 Stochastic matrices Definition 5 A square matrix p p p 3 p n p p p 3 p n p 3 p 3 p 33 p 3n P of size n n p n p n p n3 p nn is called a stochastic matrix if we have 0 p ij and sum of all the entries in a column is For example, p + p + p p n Remark We will not emphasize on this topic of stochastic matrices very much in this course We encounter such matrices in probability theory and statistics Reading Assignment: Read Textbook, Example, p and the discussion preceding this 5 Cryptography A cryptogram is a message written accoring to a secret code We describle a method of using matrix multiplication to encode and decode
44 84 CHAPTER MATRICES First, we assign a number to each letter in the letter as follows: 0 space A B 3 C 4 D 5 E 6 F G 8 H 9 I 0 J K L 3 M 4 N 5 O 6 P Q 8 R 9 S 0 T U V 3 W 4 X 5 Y 6 Z Example Write uncoded matrices of size 3 for the message LINEAR ALGEBRA Solution: L I N 9 4 E A R 5 8 A L 0 G E B 5 R A Encoding Following is the method of encoding and decoing a message: Given a message, first it is written as sequence of row matrices of a fixed size This was done in the above example, by writting the message in a sequence of uncoded matrices of size 3 These will be called uncoded matrices The message is encoded by using a square matrix A of appropriate size and simply multiplying the uncoded matrices by A 3 Decoding of the encoded matrices is done by multiplying the encoded matrices by the inverse A of A Exercise 5 (Ex 6, p3) Use the endoing matrix A
45 5 APPLICATION OF MATRIX OPERATIONS 85 to encode the message: PLEASE SEND MONEY Solution: We first write the message in uncoded row matrices of size 3 as follows: P L E 6 5 A S E 9 5 M O N So, the encoded matrices are given by: S E E Y uncoded A encoded 6 5 A A A A A A Exercise 53 (Ex p 3) Let 3 4 A be the encoding matrix Decode the cryptogram: N D Solution: We use TI to find: A
46 86 CHAPTER MATRICES So, the uncoded matrices (and corresponding letters) are given by: encoded A unencoded word A 8 HAV A 5 0 E A 6 6 A 0 8 GR 95 8 A 5 0 EAT 0 38 A W E A 5 5 EKE A ND So the message is HAVE A GREAT WEEKEND
MATRIX 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 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 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 informationLecture 2 Matrix Operations
Lecture 2 Matrix Operations transpose, sum & difference, scalar multiplication matrix multiplication, matrixvector product matrix inverse 2 1 Matrix transpose transpose of m n matrix A, denoted A T or
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 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 information1 Introduction to Matrices
1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns
More informationThe Inverse of a Matrix
The Inverse of a Matrix 7.4 Introduction In number arithmetic every number a ( 0) has a reciprocal b written as a or such that a ba = ab =. Some, but not all, square matrices have inverses. If a square
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 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 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 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 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 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 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 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 informationMatrix Algebra. Some Basic Matrix Laws. Before reading the text or the following notes glance at the following list of basic matrix algebra laws.
Matrix Algebra A. Doerr Before reading the text or the following notes glance at the following list of basic matrix algebra laws. Some Basic Matrix Laws Assume the orders of the matrices are such that
More information4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns
L. Vandenberghe EE133A (Spring 2016) 4. Matrix inverses left and right inverse linear independence nonsingular matrices matrices with linearly independent columns matrices with linearly independent rows
More informationChapter 7. Matrices. Definition. An m n matrix is an array of numbers set out in m rows and n columns. Examples. ( 1 1 5 2 0 6
Chapter 7 Matrices Definition An m n matrix is an array of numbers set out in m rows and n columns Examples (i ( 1 1 5 2 0 6 has 2 rows and 3 columns and so it is a 2 3 matrix (ii 1 0 7 1 2 3 3 1 is a
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 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 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 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 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 informationSECTION 8.3: THE INVERSE OF A SQUARE MATRIX
(Section 8.3: The Inverse of a Square Matrix) 8.47 SECTION 8.3: THE INVERSE OF A SQUARE MATRIX PART A: (REVIEW) THE INVERSE OF A REAL NUMBER If a is a nonzero real number, then aa 1 = a 1 a = 1. a 1, or
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 informationLinear Algebra Notes for Marsden and Tromba Vector Calculus
Linear Algebra Notes for Marsden and Tromba Vector Calculus ndimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of
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 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 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 informationSolution to Homework 2
Solution to Homework 2 Olena Bormashenko September 23, 2011 Section 1.4: 1(a)(b)(i)(k), 4, 5, 14; Section 1.5: 1(a)(b)(c)(d)(e)(n), 2(a)(c), 13, 16, 17, 18, 27 Section 1.4 1. Compute the following, if
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 informationMATH10212 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 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 informationIntroduction to Matrix Algebra I
Appendix A Introduction to Matrix Algebra I Today we will begin the course with a discussion of matrix algebra. Why are we studying this? We will use matrix algebra to derive the linear regression model
More informationEC9A0: Presessional Advanced Mathematics Course
University of Warwick, EC9A0: Presessional Advanced Mathematics Course Peter J. Hammond & Pablo F. Beker 1 of 55 EC9A0: Presessional Advanced Mathematics Course Slides 1: Matrix Algebra Peter J. Hammond
More informationCofactor Expansion: Cramer s Rule
Cofactor Expansion: Cramer s Rule MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Introduction Today we will focus on developing: an efficient method for calculating
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 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 informationWe know a formula for and some properties of the determinant. Now we see how the determinant can be used.
Cramer s rule, inverse matrix, and volume We know a formula for and some properties of the determinant. Now we see how the determinant can be used. Formula for A We know: a b d b =. c d ad bc c a Can we
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 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 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 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 informationHelpsheet. Giblin Eunson Library MATRIX ALGEBRA. library.unimelb.edu.au/libraries/bee. Use this sheet to help you:
Helpsheet Giblin Eunson Library ATRIX ALGEBRA Use this sheet to help you: Understand the basic concepts and definitions of matrix algebra Express a set of linear equations in matrix notation Evaluate determinants
More informationLinear Algebra: Matrices
B Linear Algebra: Matrices B 1 Appendix B: LINEAR ALGEBRA: MATRICES TABLE OF CONTENTS Page B.1. Matrices B 3 B.1.1. Concept................... B 3 B.1.2. Real and Complex Matrices............ B 3 B.1.3.
More informationMatrix Norms. Tom Lyche. September 28, Centre of Mathematics for Applications, Department of Informatics, University of Oslo
Matrix Norms Tom Lyche Centre of Mathematics for Applications, Department of Informatics, University of Oslo September 28, 2009 Matrix Norms We consider matrix norms on (C m,n, C). All results holds for
More informationL12. Special Matrix Operations: Permutations, Transpose, Inverse, Augmentation 12 Aug 2014
L12. Special Matrix Operations: Permutations, Transpose, Inverse, Augmentation 12 Aug 2014 Unfortunately, no one can be told what the Matrix is. You have to see it for yourself.  Morpheus Primary concepts:
More informationIntroduction to Matrix Algebra
Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra  1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary
More information7.4. The Inverse of a Matrix. Introduction. Prerequisites. Learning Style. Learning Outcomes
The Inverse of a Matrix 7.4 Introduction In number arithmetic every number a 0 has a reciprocal b written as a or such that a ba = ab =. Similarly a square matrix A may have an inverse B = A where AB =
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 informationMathQuest: Linear Algebra. 1. Which of the following matrices does not have an inverse?
MathQuest: Linear Algebra Matrix Inverses 1. Which of the following matrices does not have an inverse? 1 2 (a) 3 4 2 2 (b) 4 4 1 (c) 3 4 (d) 2 (e) More than one of the above do not have inverses. (f) All
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 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 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 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 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 informationNOTES on LINEAR ALGEBRA 1
School of Economics, Management and Statistics University of Bologna Academic Year 205/6 NOTES on LINEAR ALGEBRA for the students of Stats and Maths This is a modified version of the notes by Prof Laura
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 information5.3 Determinants and Cramer s Rule
290 5.3 Determinants and Cramer s Rule Unique Solution of a 2 2 System The 2 2 system (1) ax + by = e, cx + dy = f, has a unique solution provided = ad bc is nonzero, in which case the solution is given
More informationThe Singular Value Decomposition in Symmetric (Löwdin) Orthogonalization and Data Compression
The Singular Value Decomposition in Symmetric (Löwdin) Orthogonalization and Data Compression The SVD is the most generally applicable of the orthogonaldiagonalorthogonal type matrix decompositions Every
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 informationMath 115A HW4 Solutions University of California, Los Angeles. 5 2i 6 + 4i. (5 2i)7i (6 + 4i)( 3 + i) = 35i + 14 ( 22 6i) = 36 + 41i.
Math 5A HW4 Solutions September 5, 202 University of California, Los Angeles Problem 4..3b Calculate the determinant, 5 2i 6 + 4i 3 + i 7i Solution: The textbook s instructions give us, (5 2i)7i (6 + 4i)(
More informationT ( a i x i ) = a i T (x i ).
Chapter 2 Defn 1. (p. 65) Let V and W be vector spaces (over F ). We call a function T : V W a linear transformation form V to W if, for all x, y V and c F, we have (a) T (x + y) = T (x) + T (y) and (b)
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 information13 MATH FACTS 101. 2 a = 1. 7. The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions.
3 MATH FACTS 0 3 MATH FACTS 3. Vectors 3.. Definition We use the overhead arrow to denote a column vector, i.e., a linear segment with a direction. For example, in threespace, we write a vector in terms
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 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 informationMatrices Worksheet. Adding the results together, using the matrices, gives
Matrices Worksheet This worksheet is designed to help you increase your confidence in handling MATRICES. This worksheet contains both theory and exercises which cover. Introduction. Order, Addition and
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 informationDiagonalisation. Chapter 3. Introduction. Eigenvalues and eigenvectors. Reading. Definitions
Chapter 3 Diagonalisation Eigenvalues and eigenvectors, diagonalisation of a matrix, orthogonal diagonalisation fo symmetric matrices Reading As in the previous chapter, there is no specific essential
More informationLecture Notes 1: Matrix Algebra Part B: Determinants and Inverses
University of Warwick, EC9A0 Maths for Economists Peter J. Hammond 1 of 57 Lecture Notes 1: Matrix Algebra Part B: Determinants and Inverses Peter J. Hammond email: p.j.hammond@warwick.ac.uk Autumn 2012,
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 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 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 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 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 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 informationTypical Linear Equation Set and Corresponding Matrices
EWE: Engineering With Excel Larsen Page 1 4. Matrix Operations in Excel. Matrix Manipulations: Vectors, Matrices, and Arrays. How Excel Handles Matrix Math. Basic Matrix Operations. Solving Systems of
More informationLecture 4: Partitioned Matrices and Determinants
Lecture 4: Partitioned Matrices and Determinants 1 Elementary row operations Recall the elementary operations on the rows of a matrix, equivalent to premultiplying by an elementary matrix E: (1) multiplying
More informationSection 2.1. Section 2.2. Exercise 6: We have to compute the product AB in two ways, where , B =. 2 1 3 5 A =
Section 2.1 Exercise 6: We have to compute the product AB in two ways, where 4 2 A = 3 0 1 3, B =. 2 1 3 5 Solution 1. Let b 1 = (1, 2) and b 2 = (3, 1) be the columns of B. Then Ab 1 = (0, 3, 13) and
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 informationI. GROUPS: BASIC DEFINITIONS AND EXAMPLES
I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called
More informationA linear combination is a sum of scalars times quantities. Such expressions arise quite frequently and have the form
Section 1.3 Matrix Products A linear combination is a sum of scalars times quantities. Such expressions arise quite frequently and have the form (scalar #1)(quantity #1) + (scalar #2)(quantity #2) +...
More informationIntroduction to Matrices for Engineers
Introduction to Matrices for Engineers C.T.J. Dodson, School of Mathematics, Manchester Universit 1 What is a Matrix? A matrix is a rectangular arra of elements, usuall numbers, e.g. 1 08 4 01 1 0 11
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 information15.062 Data Mining: Algorithms and Applications Matrix Math Review
.6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop
More information6. Cholesky factorization
6. Cholesky factorization EE103 (Fall 201112) triangular matrices forward and backward substitution the Cholesky factorization solving Ax = b with A positive definite inverse of a positive definite matrix
More informationChapter 17. Orthogonal Matrices and Symmetries of Space
Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length
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 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 informationThe Characteristic Polynomial
Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem
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 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 informationThe Determinant: a Means to Calculate Volume
The Determinant: a Means to Calculate Volume Bo Peng August 20, 2007 Abstract This paper gives a definition of the determinant and lists many of its wellknown properties Volumes of parallelepipeds are
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 informationThe basic unit in matrix algebra is a matrix, generally expressed as: a 11 a 12. a 13 A = a 21 a 22 a 23
(copyright by Scott M Lynch, February 2003) Brief Matrix Algebra Review (Soc 504) Matrix algebra is a form of mathematics that allows compact notation for, and mathematical manipulation of, highdimensional
More informationChapter 6. Orthogonality
6.3 Orthogonal Matrices 1 Chapter 6. Orthogonality 6.3 Orthogonal Matrices Definition 6.4. An n n matrix A is orthogonal if A T A = I. Note. We will see that the columns of an orthogonal matrix must be
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 informationEigenvalues, Eigenvectors, Matrix Factoring, and Principal Components
Eigenvalues, Eigenvectors, Matrix Factoring, and Principal Components The eigenvalues and eigenvectors of a square matrix play a key role in some important operations in statistics. In particular, they
More information