Matrix representation of a linear map

Transcription

1 Matrix representation of a linear map As before, let e i = (0,..., 0, 1, 0,..., 0) T, with 1 in the i th place and 0 elsewhere, be standard basis vectors. Given linear map f : R n R m we get n column vectors (of size m) viz., f (e 1 ),..., f (e n ). Place them side by side., Thus if f (e j ) = (f 1j, f 2j,..., f mj ) T, then we obtain an array of the form M f = f 11 f f 1n f 21 f f 2n.. f m1 f m2... f mn. Such an array is called a matrix with m rows and n columns or a matrix of size m n. In short, we write M f = (f ij ). We call M f to be the matrix associated with the linear map f : R n R m. 12/45

2 13/45 If we let Mat m n (R), or simply, M m,n, denote the set of all m n matrices with entries in R, then we obtain: M : L(n, m) M m,n f M f The map M is a one-to-one correspondence (i.e., one-one and onto map) between the set of all linear maps from R n to R m and the set of all m n matrices with entries in R. Note: Two matrices are equal if their sizes are the same and corresponding entries are the same. M is one-one (or injective), means M f = M g for some f, g L(n, m) f = g, whereas M is onto (or surjective) means that A Mat m n (R) A = M f for some f L(n, m).

3 13/45 If we let Mat m n (R), or simply, M m,n, denote the set of all m n matrices with entries in R, then we obtain: M : L(n, m) M m,n f M f The map M is a one-to-one correspondence (i.e., one-one and onto map) between the set of all linear maps from R n to R m and the set of all m n matrices with entries in R. Note: Two matrices are equal if their sizes are the same and corresponding entries are the same. M is one-one (or injective), means M f = M g for some f, g L(n, m) f = g, whereas M is onto (or surjective) means that A Mat m n (R) A = M f for some f L(n, m).

4 13/45 If we let Mat m n (R), or simply, M m,n, denote the set of all m n matrices with entries in R, then we obtain: M : L(n, m) M m,n f M f The map M is a one-to-one correspondence (i.e., one-one and onto map) between the set of all linear maps from R n to R m and the set of all m n matrices with entries in R. Note: Two matrices are equal if their sizes are the same and corresponding entries are the same. M is one-one (or injective), means M f = M g for some f, g L(n, m) f = g, whereas M is onto (or surjective) means that A Mat m n (R) A = M f for some f L(n, m).

5 13/45 If we let Mat m n (R), or simply, M m,n, denote the set of all m n matrices with entries in R, then we obtain: M : L(n, m) M m,n f M f The map M is a one-to-one correspondence (i.e., one-one and onto map) between the set of all linear maps from R n to R m and the set of all m n matrices with entries in R. Note: Two matrices are equal if their sizes are the same and corresponding entries are the same. M is one-one (or injective), means M f = M g for some f, g L(n, m) f = g, whereas M is onto (or surjective) means that A Mat m n (R) A = M f for some f L(n, m).

6 If we let Mat m n (R), or simply, M m,n, denote the set of all m n matrices with entries in R, then we obtain: M : L(n, m) M m,n f M f The map M is a one-to-one correspondence (i.e., one-one and onto map) between the set of all linear maps from R n to R m and the set of all m n matrices with entries in R. Note: Two matrices are equal if their sizes are the same and corresponding entries are the same. M is one-one (or injective), means M f = M g for some f, g L(n, m) f = g, whereas M is onto (or surjective) means that A Mat m n (R) A = M f for some f L(n, m). M preserves addition and scalar multiplication, i.e., M f +g = M f +M g and M αf = αm f, f, g L(n, m), α R. 13/45

7 Examples: 14/45

8 14/45 Examples: 1. For the identity map id : R n R n, M id = I = I n :=

9 14/45 Examples: 1. For the identity map id : R n R n, M id = I = I n :=.. = (δ ij) wehre δ ij is the Kronecker delta defined by δ ij = 1 if i = j and δ ij = 0, if i j.

10 14/45 Examples: 1. For the identity map id : R n R n, M id = I = I n :=.. = (δ ij) wehre δ ij is the Kronecker delta defined by δ ij = 1 if i = j and δ ij = 0, if i j. 2. The matrix of the linear map T : R 2 R 2 which interchanges coordinates, is

11 14/45 Examples: 1. For the identity map id : R n R n, M id = I = I n :=.. = (δ ij) wehre δ ij is the Kronecker delta defined by δ ij = 1 if i = j and δ ij = 0, if i j. 2. The matrix of the linear map T : R 2 ( R 2 which ) 0 1 interchanges coordinates, is M T =. 1 0

12 14/45 Examples: 1. For the identity map id : R n R n, M id = I = I n :=.. = (δ ij) wehre δ ij is the Kronecker delta defined by δ ij = 1 if i = j and δ ij = 0, if i j. 2. The matrix of the linear map T : R 2 ( R 2 which ) 0 1 interchanges coordinates, is M T = For the linear map µ α : R n R n corresponding to multiplication by a fixed scalar α, the associated matrix is

13 14/45 Examples: 1. For the identity map id : R n R n, M id = I = I n :=.. = (δ ij) wehre δ ij is the Kronecker delta defined by δ ij = 1 if i = j and δ ij = 0, if i j. 2. The matrix of the linear map T : R 2 ( R 2 which ) 0 1 interchanges coordinates, is M T = For the linear map µ α : R n R n corresponding to multiplication by a fixed scalar α, the associated matrix is the scalar matrix diag(α,..., α) := (αδ ij )

14 15/45 4. Rotation of the plane by an angle θ defines a linear map R θ : R 2 R 2 (Linearity of R θ can be shown by the law of congruent triangles or by finding a formula for R θ (x, y) using polar coordinates and trigonometric identities). The matrix of R θ is about the ( ) cos θ sin θ M Rθ = sin θ cos θ

15 15/45 4. Rotation of the plane by an angle θ defines a linear map R θ : R 2 R 2 (Linearity of R θ can be shown by the law of congruent triangles or by finding a formula for R θ (x, y) using polar coordinates and trigonometric identities). The matrix of R θ is about the ( ) cos θ sin θ M Rθ = sin θ cos θ 5. The inclusion map R 2 R 3 that sends (x, y) R 2 to (x, y, 0) R 3 is clearly a linear map and its matrix is

16 15/45 4. Rotation of the plane by an angle θ defines a linear map R θ : R 2 R 2 (Linearity of R θ can be shown by the law of congruent triangles or by finding a formula for R θ (x, y) using polar coordinates and trigonometric identities). The matrix of R θ is about the ( ) cos θ sin θ M Rθ = sin θ cos θ 5. The inclusion map R 2 R 3 that sends (x, y) R 2 to (x, y, 0) R 3 is clearly a linear map and its matrix is In a similar way, we can write down the (n + t) n matrix corresponding to the inclusion map R n R n+t.

17 Operations on matrices 16/45 Set of all matrices of size m n is denoted by M m,n

18 Operations on matrices 16/45 Set of all matrices of size m n is denoted by M m,n As before, for A = (a ij ), B = (b ij ) M m,n and α R, A + B := (a ij + b ij ); αa := (αa ij ) Matrix addition and scalar multiplicaion satisfies the usual properties. The zero element is the zero matrix 0, all of whose entries are 0 and the negative of a matrix A = (a ij ) is the matrix A = ( a ij ).

19 Operations on matrices 16/45 Set of all matrices of size m n is denoted by M m,n As before, for A = (a ij ), B = (b ij ) M m,n and α R, A + B := (a ij + b ij ); αa := (αa ij ) Matrix addition and scalar multiplicaion satisfies the usual properties. The zero element is the zero matrix 0, all of whose entries are 0 and the negative of a matrix A = (a ij ) is the matrix A = ( a ij ).

20 Operations on matrices 16/45 Set of all matrices of size m n is denoted by M m,n As before, for A = (a ij ), B = (b ij ) M m,n and α R, A + B := (a ij + b ij ); αa := (αa ij ) Matrix addition and scalar multiplicaion satisfies the usual properties. The zero element is the zero matrix 0, all of whose entries are 0 and the negative of a matrix A = (a ij ) is the matrix A = ( a ij ). Transpose operation introduced earlier can be extended to all matrices A T := (b ij ) M n,m where b ij := a ji Note that (αa + βb) T = αa T + βb T for any α, β R.

21 Operations on matrices 16/45 Set of all matrices of size m n is denoted by M m,n As before, for A = (a ij ), B = (b ij ) M m,n and α R, A + B := (a ij + b ij ); αa := (αa ij ) Matrix addition and scalar multiplicaion satisfies the usual properties. The zero element is the zero matrix 0, all of whose entries are 0 and the negative of a matrix A = (a ij ) is the matrix A = ( a ij ). Transpose operation introduced earlier can be extended to all matrices A T := (b ij ) M n,m where b ij := a ji Note that (αa + βb) T = αa T + βb T for any α, β R.

22 Operations on matrices Set of all matrices of size m n is denoted by M m,n As before, for A = (a ij ), B = (b ij ) M m,n and α R, A + B := (a ij + b ij ); αa := (αa ij ) Matrix addition and scalar multiplicaion satisfies the usual properties. The zero element is the zero matrix 0, all of whose entries are 0 and the negative of a matrix A = (a ij ) is the matrix A = ( a ij ). Transpose operation introduced earlier can be extended to all matrices A T := (b ij ) M n,m where b ij := a ji Note that (αa + βb) T = αa T + βb T for any α, β R. Products of linear maps is not linear (in fact not even defined), in general. But composites of linear maps are linear. How does this relate to matrices? 16/45

23 17/45 Composition of Linear Maps and Matrix Multiplication Question: Let f : R n R m, g : R p R n be linear maps. If A := M f and B := M g then M f g =?

24 Composition of Linear Maps and Matrix Multiplication Question: Let f : R n R m, g : R p R n be linear maps. If A := M f and B := M g then M f g =? f g is a map of R p to R m. For 1 j p, ( n ) (f g)(e j ) = f (g(e j )) = f b kj e k = = n b kj f (e k ) = k=1 k=1 ( n m ) b kj a ik e i k=1 ( m n ) a ik b kj e i i=1 k=1 Thus M f g = C, where C = (c ij ) is the m p matrix whose (i, j) th entry is c ij = n k=1 a ikb kj. We call C the product of the m n matrix A and the n p matrix B, and write C = AB. i=1 17/45

25 Basic Properties of Matrix Multiplication 18/45 The following properties are readily derived from the corresponding properties of linear maps and the one-to -one correspondence between linear maps and matrices. Assume that the matrices A, B, C are of appropriate sizes so that all relevant products are defined. 1. Associativity: A(BC) = (AB)C.

26 Basic Properties of Matrix Multiplication 18/45 The following properties are readily derived from the corresponding properties of linear maps and the one-to -one correspondence between linear maps and matrices. Assume that the matrices A, B, C are of appropriate sizes so that all relevant products are defined. 1. Associativity: A(BC) = (AB)C. 2. Right and Left Distributivity: A(B + C) = AB + AC, (B + C)A = BA + CA

27 Basic Properties of Matrix Multiplication The following properties are readily derived from the corresponding properties of linear maps and the one-to -one correspondence between linear maps and matrices. Assume that the matrices A, B, C are of appropriate sizes so that all relevant products are defined. 1. Associativity: A(BC) = (AB)C. 2. Right and Left Distributivity: A(B + C) = AB + AC, (B + C)A = BA + CA 3. Multiplicative identity: For any A M m,n, AI n = A and I m A = A In particular, if A is a n n matrix, then AI n = A = I n A. Remark: Matrix multiplication is not commutative in general, i.e., AB need not equal BA even when both the products are defined. Exercise: Find examples! 18/45

28 Basic Properties of Matrix Multiplication The following properties are readily derived from the corresponding properties of linear maps and the one-to -one correspondence between linear maps and matrices. Assume that the matrices A, B, C are of appropriate sizes so that all relevant products are defined. 1. Associativity: A(BC) = (AB)C. 2. Right and Left Distributivity: A(B + C) = AB + AC, (B + C)A = BA + CA 3. Multiplicative identity: For any A M m,n, AI n = A and I m A = A In particular, if A is a n n matrix, then AI n = A = I n A. 4. Transpose of Product: (AB) T = B T A T. Remark: Matrix multiplication is not commutative in general, i.e., AB need not equal BA even when both the products are defined. Exercise: Find examples! 18/45

29 Invertible Maps 19/45 Recall the set-theoretic definition of invertible functions. Let X, Y be sets. A function f : X Y is said to be invertible if there exists g : Y X satisfying g f = id X and f g = id Y. In this case the function g is unique (check!) and it is called the inverse of f and denoted by f 1.

30 Invertible Maps Recall the set-theoretic definition of invertible functions. Let X, Y be sets. A function f : X Y is said to be invertible if there exists g : Y X satisfying g f = id X and f g = id Y. In this case the function g is unique (check!) and it is called the inverse of f and denoted by f 1. For any function f : X Y, f is invertible f is one-one and onto. If f : R n R n is an invertible linear map, then f 1 is linear. [To see this, let y 1, y 2 R n. Then there are x 1, x 2 R n such that f (x i ) = y i, i.e., x i = f 1 (y i ) for i = 1, 2. Thus f 1 (y 1 + y 2 ) = f 1 (f (x 1 ) + f (x 2 )) and using the linearity of f, we find that this is equal to f 1 (f (x 1 + x 2 )) = x 1 + x 2 = f 1 (y 1 ) + f 1 (y 2 ). Similarly f 1 preserves scalar multiplication. 19/45