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Draft. Do not cite or quote. Similarity We start with a linear transformation L : V → W and with bases X = x1 , . . . , xn of V and Y = y1 , . . . , ym of W . Recall that every vector v ∈ V can be written in exactly one way as a linear combination of the basis vectors of X: v = c1 x1 + · · · + cn xn , where all ci ∈ R; we call the n × 1 column vector (c1 , . . . , cn )> the coordinate list of v wrt X (with respect to X). Of course, every vector w ∈ W has a coordinate list wrt Y . Linear transformations defined on Rn are easy to describe. Theorem 0.1 If L : Rn → Rm is a linear transformation, then there exists a unique m × n matrix A such that L(y) = Ay for all y ∈ Rn (here, y is an n × 1 column matrix and Ay is matrix multiplication). Proof. If e1 , . . . , en is the standard basis of Rn and e01 , . . . , e0m is the standard basis of Rm , define A = [aij ] to be the matrix whose jth column is the coordinate list of L(ej ). If S : Rn → Rm is defined by P S(y) = Ay, then S = L because both agree on a basis: L(ej ) = i aij ei = Aej . Uniqueness of A follows from the fact that if L(y) = By for all y, then Bej = L(ej ) = Aej for all j; that is, the columns of A and B are the same. Theorem 0.1 establishes the connection between linear transformations and matrices, and the definition of matrix multiplication arises from applying this construction to the composite of two linear transformations. Definition Let X = v1 , . . . , vn be a basis of V and let Y = w1 , . . . , wm be a basis of W . If L : V → W is a linear transformation, then the matrix of L is the m × n matrix A = [aij ] whose jth column a1j , a2j , . . . , amj is the coordinate list of L(vj ) wrt Y : L(vj ) = P m i=1 aij wi = a1j w1 + a2j w2 + · · · + amj wm . Since the matrix A depends on the choice of bases X and Y , we will write A = Y [L]X Abstract Algebra for Teachers Draft. c 2011 1 Draft. Do not cite or quote. when it is necessary to display them. Varying any of them most likely changes the matrix. Consider the important special case LA : Rn → Rm given by LA (y) = Ay, where A is an m × n matrix and y is an n × 1 column vector. If E = e1 , . . . , en and E 0 = e01 , . . . , e0m are the standard bases of Rn and Rm , respectively, then the definition of matrix multiplication says that LA (ej ) = Aej is the jth column of A. But Aej = a1j e01 + a2j e02 + · · · + amj e0m ; that is, the coordinates of LA (ej ) = Aej with respect to the basis e01 , . . . , e0m are (a1j , . . . , amj ). Therefore, the matrix associated to LA is the original matrix A: E 0 [LA ]E = A. In case V = W , we often let the bases X = v1 , . . . , vn and Y = w1 , . . . , wm coincide. If 1V : V → V , given by v 7→ v, is the identity linear transformation, then X [1V ]X is the n × n identity matrix I, which has 1’s on the diagonal and 0’s elsewhere. On the other hand, if X and Y are different bases, then Y [1V ]X is not the identity matrix. The matrix Y [1V ]X is called the transition matrix from X to Y ; its columns are the coordinate lists of the v’s with respect to the w’s. Example 1. Let X = e1 , e2 = (1, 0), (0, 1) be the standard basis of R2 . If L : R2 → R2 is rotation by 90◦ , then L : e1 7→ e2 and e2 7→ −e1 . Hence, the matrix of L relative to X is " # 0 −1 . X [L]X = 1 0 If we re-order X to obtain the new basis Y = η1 , η2 , where η1 = e2 and η2 = e1 , then L(η1 ) = L(e2 ) = −e1 = −η2 and L(η2 ) = L(e1 ) = e2 = η1 . The matrix of L relative to Y is " # 0 1 . Y [L]Y = −1 0 2. Let L : R2 → R2 be rotation by α degrees about the origin. Take X = Y to be the standard basis e1 = (1, 0), e2 = (0, 1). Now L(e1 ) = (cos α, sin α) and L(e2 ) = (cos(90 + α), sin(90 + α) = (− sin α, cos α). Thus the matrix of L wrt standard basis is " # cos α − sin α . X [L]X = sin α cos α 2 Abstract Algebra for Teachers Draft. c 2011 Draft. Do not cite or quote. 3. If L : C → C is complex conjugation, and if we choose the bases X = Y = 1, i, then the matrix is " # 1 0 . X [L]X = 0 −1 The next theorem shows where the definition of matrix multiplication comes from: the product of two matrices is the matrix of a composite. Theorem 0.2 Let L : V → W and S : W → U be linear transformations. Choose bases X = x1 , . . . , xn of V , Y = y1 , . . . , ym of W , and Z = z1 , . . . , z` of U . Then Z [S ◦ L]X = Z [S]Y Y [L]X , where the product on the right is matrix multiplication. Proof. Let Y [L]X = [aij ], so P that L(xj ) = Z [S]Y = [bqp ], so that S(yp ) = q bqp zq . Then X S ◦ L(xj ) = S(L(xj )) = S apj yp P p apj yp , and let p = X apj S(yp ) = XX p where cqj = P p bqp apj . Z [S p apj bqp zq = q X cqj zq , q Therefore, ◦ L]X = [cqj ] = Z [S]Y Y [L]X . Corollary 0.3 Matrix multiplication is associative. Proof. Let A be an m × n matrix, let B be an n × p matrix, and let C be a p × q matrix. There are linear transformations L L L C B A Rq → Rp → Rn → Rm We know C = [LC ], B = [LB ], and A = [LA ]. Then [LA ◦ (LB ◦ L)] = [LA ][LB ◦ LC ] = [LA ]([LB ][LC ]) = A(BC). Abstract Algebra for Teachers Draft. c 2011 3 Draft. Do not cite or quote. On the other hand, [(LA ◦ LB ) ◦ LC ] = [LA ◦ LB ][LC ] = ([LA ][LB ])[LC ] = (AB)C. Since composition of functions is associative, LA ◦ (LB ◦ LC ) = (LA ◦ LB ) ◦ LC , and so A(BC) = [LA ◦ (LB ◦ LC )] = [(LA ◦ LB ) ◦ LC ] = (AB)C. The connection with composition of linear transformations is the real reason why matrix multiplication is associative. Corollary 0.4 Let L : V → W be a linear transformation of vector spaces V and W , and let X and Y be bases of V and W , respectively. If there is a linear transformation T : W → V with L ◦ T = 1V , then the matrix of T is the inverse of the matrix of L : X [T ]Y = (Y [L]X )−1 . Proof. We have I = Y [1W ]Y = Y [L]X . X [1V ]X = X [T ]Y Y [L]X X [T ]Y , and so I = The next corollary determines all the matrices arising from the same linear transformation as we vary bases. Here we take the same basis fore and aft: X = Y . Corollary 0.5 Let L : V → V be a linear transformation on a vector space V . If X and Y are bases of V , then there is a nonsingular matrix P (namely, the transition matrix P = Y [1V ]X ) with entries in R so that −1 . Y [L]Y = P X [L]X P Conversely, if B = P AP −1 , where B, A, and P are n × n matrices with P nonsingular, then there is a linear transformation L : Rn → Rn and bases X and Y of Rn such that B = Y [L]Y and A = X [L]X . Proof. The first statement follows from Theorem ?? and associativity: Y [L]Y = Y [1V L1V ]Y = (Y [1V ]X )(X [L]X )(X [1V ]Y ). Set P = Y [1V ]X and note that Corollary ?? gives P −1 = X [1V ]Y . 4 Abstract Algebra for Teachers Draft. c 2011 Draft. Do not cite or quote. The converse is a bit longer. Definition Two n × n matrices B and A with entries in R are similar if there is a nonsingular matrix P with entries in R such that B = P AP −1 . Thus, two matrices arise from the same linear transformation on a vector space V (from different choices of bases) if and only if they are similar. Abstract Algebra for Teachers Draft. c 2011 5