Math 4377/6308 Advanced Linear Algebra


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1 2.3 Composition Math 4377/6308 Advanced Linear Algebra 2.3 Composition of Linear Transformations Jiwen He Department of Mathematics, University of Houston math.uh.edu/ jiwenhe/math4377 Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, / 24
2 2.3 Composition of Linear Transformations Compositions of Maps Basic Properties of Compositions Multiplication of Matrices Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, / 24
3 Composition of Linear Transformations Theorem (2.9) Let V, W, Z be vector spaces over a field F, and T : V W, U : W Z be linear. Then UT : V Z is linear. Theorem (2.10) Let V be a vector space and T, U 1, U 2 L(V ). Then (a) T (U 1 + U 2 ) = TU 1 + TU 2 and (U 1 + U 2 )T = U 1 T + U 2 T. (b) T (U 1 U 2 ) = (TU 1 )U 2. (c) TI = IT = T. (d) a(u 1 U 2 ) = (au 1 )U 2 = U 1 (au 2 ) for all scalars a. Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, / 24
4 Matrix Multiplication Let T : V W, U : W Z be linear, α = {v 1,, v n }, β = {w 1,, w m }, γ = {z 1,, z p } ordered bases for V, W, Z, and A = [U] γ β, B = [T ]β α. Consider [UT ] γ α: ( m ) (UT )(v j ) = U(T (v j )) = U B kj w k = k=1 m B kj U(w k ) k=1 ( m p ) = B kj A ik z i = k=1 i=1 ( p m ) A ik B kj z i i=1 k=1 Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, / 24
5 Matrix Multiplication (cont.) Definition Let A, B be m n, n p matrices. The product AB is the m p matrix with (AB) ij = n A ik B kj, for 1 i m, 1 j p. k=1 Theorem (2.11) Let V, W, Z be finitedimensional vector spaces with ordered bases α, β, γ, and T : V W, U : W Z be linear. Then [UT ] γ α = [U] γ β [T ]β α Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, / 24
6 Matrix Multiplication (cont.) Corollary Let V be a finitedimensional vector space with ordered basis β, and T, U L(V ). Then [UT ] β = [U] β [T ] β. Definition The Kronecker delta is defined by δ ij = 1 if i = j and δ ij = 0 if i j. The n n identity matrix I n is defined by (I n ) ij = δ ij. Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, / 24
7 Matrix Notation 2.3 Composition Composition Addition Multiplication Power Matrix Notation Two ways to denote m n matrix A: 1 In terms of the columns of A: A = [ a 1 a 2 a n ] 2 In terms of the entries of A: a 11 a 1j a 1n.. A = a i1 a ij a in... a m1 a mj a mn Main diagonal entries: Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, / 24
8 Matrix Addition: Theorem Theorem (Addition) Let A, B, and C be matrices of the same size, and let r and s be scalars. Then a. A + B = B + A d. r(a + B) = ra + rb b. (A + B) + C = A + (B + C) e. (r + s) A = ra + sa c. A + 0 = A f. r (sa) = (rs) A Zero Matrix 0 = Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, / 24
9 Matrix Multiplication Matrix Multiplication Multiplying B and x transforms x into the vector Bx. we multiply A and Bx, we transform Bx into A (Bx). the composition of two mappings. In turn, if So A (Bx) is Define the product AB so that A (Bx) = (AB) x. Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, / 24
10 Matrix Multiplication: Definition Suppose A is m n and B is n p where x 1 x 2 B = [b 1 b 2 b p ] and x =.. Then Bx =x 1 b 1 + x 2 b x p b p x p and A (Bx) =A(x 1 b 1 + x 2 b x p b p ) = A (x 1 b 1 ) + A (x 2 b 2 ) + + A (x p b p ) = x 1 Ab 1 + x 2 Ab x p Ab p = [Ab 1 Ab 2 Ab p ] x 1 x 2. x p. Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, / 24
11 Matrix Multiplication: Definition (cont.) Therefore, and by defining we have A (Bx) = (AB) x. A (Bx) = [Ab 1 Ab 2 Ab p ] x. AB = [Ab 1 Ab 2 Ab p ] Note that Ab 1 is a linear combination of the columns of A, Ab 2 is a linear combination of the columns of A, etc. Each column of AB is a linear combination of the columns of A using weights from the corresponding columns of B. Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, / 24
12 Matrix Multiplication: Example Example Compute AB where A = Solution: Ab 1 = = [ = AB = and B = [ ]. ] 4 2 [ ], Ab 2 = = Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, / 24
13 Matrix Multiplication: Example Example If A is 4 3 and B is 3 2, then what are the sizes of AB and BA? Solution: AB = = BA would be which is. If A is m n and B is n p, then AB is m p. Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, / 24
14 RowColumn Rule for Computing AB (alternate method) The definition AB = [Ab 1 Ab 2 Ab p ] is good for theoretical work. When A and B have small sizes, the following method is more efficient when working by hand. RowColumn Rule for Computing AB If AB is defined, let (AB) ij denote the entry in the ith row and jth column of AB. Then (AB) ij = a i1 b 1j + a i2 b 2j + + a in b nj, i.e., a i1 a i2 a in b 1j b 2j. = (AB) ij b nj Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, / 24
15 RowColumn Rule for Computing AB: Example Example [ A = defined ], B = Compute AB, if it is Solution: Since A is 2 3 and B is 3 2, then AB is defined and AB is. [ ] 2 3 [ ] AB = = [ ] [ ] = Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, / 24
16 RowColumn Rule for Computing AB: Example (cont.) [ ] = [ ] [ ] = [ ] So AB = [ ]. Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, / 24
17 Matrix Multiplication: Theorem Theorem (Multiplication) Let A be m n and let B and C have sizes for which the indicated sums and products are defined. a. A (BC) = (AB)C (associative law of multiplication) b. A (B + C) = AB + AC (left  distributive law) c. (B + C) A = BA + CA (rightdistributive law) d. r(ab) = (ra)b = A(rB) for any scalar r e. I m A = A = AI n (identity for matrix multiplication) Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, / 24
18 Matrix Power 2.3 Composition Composition Addition Multiplication Power Powers of A A k = A A }{{} k Example [ = ] 3 = [ [ ] [ ] [ ] = ] [ [ ] ] Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, / 24
19 Properties of Matrix Multiplication Theorem (2.12) Let A be m n matrix, B, C be n p matrices, and D, E be q m matrices. Then (a) A(B + C) = AB + AC and (D + E)A = DA + EA. (b) a(ab) = (aa)b = A(aB) for any scalar a. (c) I m A = A = AI n. (d) If V is an ndimensional vector space with ordered basis β, then [I V ] β = I n. Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, / 24
20 Properties of Matrix Multiplication (cont.) Corollary Let A be m n matrix, B 1,, B k be n p matrices, C 1,, C k be q m matrices, and a 1,, a k be scalars. Then and ( k ) A a i B i = i=1 ( k ) a i C i A = i=1 k a i AB i i=1 k a i C i A i=1 Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, / 24
21 Properties of Matrix Multiplication (cont.) Theorem (2.13) Let A be m n matrix and B be n p matrix, and u j, v j the jth columns of AB, B. Then (a) u j = Av j. (b) v j = Be j. Theorem (2.14) Let V, W be finitedimensional vector spaces with ordered bases β, γ, and T : V W be linear. Then for u V : [T (u)] γ = [T ] γ β [u] β Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, / 24
22 Leftmultiplication Transformations Definition Let A be m n matrix. The leftmultiplication transformation L A is the mapping L A : F n F m defined by L A (x) = Ax for each column vector x F n. Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, / 24
23 Leftmultiplication Transformations (cont.) Theorem (2.15) Let A be m n matrix, then L A : F n F m is linear, and if B is m n matrix and β, γ are standard ordered bases for F n, F m, then: (a) [L A ] γ β = A. (b) L A = L B if and only if A = B. (c) L A+B = L A + L B and L aa = al A for all a F. (d) For linear T : F n F m,there exists a unique m n matrix C such that T = L C, and C = [T ] γ β. (e) If E is an n p matrix, then L AE = L A L E. (f) If m = n, then L In = I F n. Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, / 24
24 Theorem (2.16) Let A, B, C be matrices such that A(BC) is defined. Then (AB)C is also defined and A(BC) = (AB)C. Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, / 24
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