n The function φβ : V ! F which sends v to [v]β is an isomorphism. Although an isomorphism doesn't mean that V is identical to F n, it does mean that coordinates work exactly the same. One place where this can be confusing is when Coordinates V is F n, but the basis β is not the standard ba- Math 130 Linear Algebra sis. Having two bases for the same vector space is D Joyce, Fall 2015 confusing, but sometimes useful. Throughout this discussion, F refers to a fixed field. In application, F will usually be R. Matrices for transformations V ! W . So far, n m We've seen how each linear transformation T : we can represent a transformation T : F ! F F n ! F m corresponds to an m × n matrix A. The by an m × n matrix A. The entries for A were determined by what T did to the standard basis of matrix A is the standard matrix representing T and n th F . The coordinates of T (ej) were placed in the its j column consists of the coordinates of the vec- th th j column of A. It followed that for any vector v tor T (ej) where ej is the j vector in the standard n th basis of F n. in F , the i coordinate of it's image w = T (v) was We'll use this connection as we derive algebraic operations on linear transformations to translate n X those operations to algebraic operations on matri- wi = Ai1v1 + Ai2v2 + ··· + Aijvn = Aijvj ces. j=1 Before we do that, however, we should see that we can use use matrices to represent linear trans- We can represent a transformation T : V ! W formations between other vector spaces so long as by a matrix as well, but it will have to be relative we've got coordinates for them. to one basis β for V and another basis γ for W . Suppose that β = (b1; b2;:::; bn) is an ordered basis for V and γ = (c ; c ;:::; c ) is an ordered Extending the standard matrix to transfor- 1 2 m basis for W . We'll put the γ-coordinates of the mations T : V ! W . Coordinates on vector T (b ) in the jth column of A, just like we did before. spaces are determined by bases for those spaces. j If we need to indicate the bases that were used to Recall that a basis β = (b ; b ;:::; b ) of a vec- γ 1 2 n create the matrix A, we'll write A = [T ] . tor space V sets up a coordinate system on V . Each β What we've actually done here was that we used vector v in V can be expressed uniquely as a linear the isomorphisms φ : V ! Rn and φ : V ! Rm combination of the basis vectors β γ to transfer the linear transformation T : V ! W to a linear transformation T 0 : Rn ! Rm. Define v = v1b1 + v2b2 + ··· + vnbn: 0 −1 T as φγ ◦ T ◦ φβ . The coefficients in that linear combination form the coordinate vector [v] relative to β T β VW- 2 3 v1 φβ φγ 6v27 [v] = 6 7 β 6 . 7 ? T 0 ? 4 . 5 Rn - Rm vn 1 Whereas T describes the linear transformation where a linear transformation T corresponds to the γ V ! W without mentioning the bases or coor- matrix A = [T ]β. The entries Aij of A describe how 0 dinates, T describes the linear transformation in to express T evaluated at the β-basis vector bj as terms of coordinates. We used those coordinates to a linear combination of the γ-basis vectors. γ describe the entries of the matrix A = [T ]β. X (bj) = Aijci γ The vector space of linear transformations We'll use this bijection φβ to define addition and HomF (V; W ). Let's use the notation HomF (V; W ) scalar multiplication on Mm×n, and when we do γ for the set of linear transformations V ! W , or that, φβ will be an isomorphism of vector spaces. more simply Hom(V; W ) when there's only one field First consider addition. Given T : V ! W and γ under consideration. S : V ! W . Let A = [T ]β be the standard matrix γ This is actually a vector space itself. If you have for T , and let B = [S]β be the standard matrix for two transformations, S : V ! W and T : V ! W , S. Then X then define their sum by T (bj) = Aijci, and i (S + T )(v) = S(v) + T (v): X S(bj) = Bijci. Therefore Check that this sum is a linear transformation by i X verifying that (T + S)(bj) = (Aij + Bij)ci: (1) for two vectors u and v in V , i Thus, the entries in the standard matrix for T + S (S + T )(u + v) = S(u + v) + S(u + v), and are sums of the corresponding entries of the stan- dard matrices for T and S. We now know how we (2) for a vector v and a scalar c, should define addition of matrices. (S + T )(cv) = c(S + T )(v): Definition 1. We define addition of two m × n matrices coordinatewise: (A + B)ij = Aij + Bij. Next, define scalar multiplication on Hom(V; W ) by Next, consider scalar multiplication. Given a (cS)(v) = c(S)(v) scalar c and a linear transformation T : V ! W , let A be the standard matrix for T . Then and show that's a linear transformation. X T (bj) = Aijci. Therefore There are still eight axioms for a vector space i that need to be checked to show that with these two X (cT )(b ) = cA c : operations Hom(V; W ) is, indeed, a vector space. j ij i i But note how the operations of addition and scalar Thus, the entries in the standard matrix for cT are multiplication were both defined by those opera- each c times the corresponding entries of the stan- tions on W . Since the axioms hold in W , they'll dard matrice for T . We now know how we should also hold in Hom(V; W ). define scalar multiplication of matrices. Definition 2. We define scalar multiplication of a The vector space of matrices Mm×n(F ). scalar c and an m × n matrix A coordinatewise: Mm×n(F ) is the set of m × n matrices with entries in the field F . We'll drop the subscript F when the (cA)ij = cAij. field is understood. With these definitions of addition and scalar mul- When V and W are endowed with bases β and tiplication, Mm×n(F ) becomes a vector space over γ ' γ, we have a bijection φβ : Hom(V; W ) ! Mm×n F , and it is isomorphic to HomF (V; W ). 2 Invariants. Note that the isomorphism ∼ Hom(V; W ) = Mm×n depends on the ordered bases you choose. With different β and γ you'll get different matrices for the same transformation. In some sense, the transformation holds intrin- sic information, while the matrix is a description which varies depending on the bases you happen to choose. When you're looking for properties of the trans- formation, they shouldn't be artifacts of the chosen bases but properties that hold for all bases. When we get to looking for properties of trans- formations, we'll make sure that they're invariant under change of basis. Math 130 Home Page at http://math.clarku.edu/~ma130/ 3.