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By S. E. Payne

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We want to figure out what is the coordinate matrix [TA ]B3 . ) We claim that [TA ]B3 = (a11 , a12 , a13 , a21 , a22 , a23 )T . Because of the order in which we listed the basis vectors fij , this is equivalent to saying that TA = i,j aij fji . If we evaluate this sum at (ek ) we get aij fji (ek ) = i,j aik fki (ek ) = i This establishes our claim. aik hi = i a1k a2k = Aek = TA (ek ). 40 CHAPTER 4. LINEAR TRANSFORMATIONS Recall that B1 = (u1 , . . , un ) is an ordered basis for U over the field F , and that B2 = (v1 , .

Un ) be an ordered basis for the vector space U over the field F , and let B2 = (v1 , . . , vm ) be an ordered basis for the vector space V over F . Let A be an m × n matrix over F . Define TA : U → V by [TA (u)]B2 = A · [u]B1 for all u ∈ U . It is quite straightforward to show that TA ∈ L(U, V ). It is also clear (by letting u = uj ), that the j th column of A is [T (uj )]B2 . Conversely, if T ∈ L(U, V ), and if we define the matrix A to be the matrix with j th column equal to [T (uj )]B2 , then [T (u)]B2 = A · [u]B1 for all u ∈ U .

Prove that ST is invertible if and only if both S and T are invertible. 13. Suppose that V is finite dimensional and T ∈ L(V ). Prove that T is a scalar multiple of the identity if and only if ST = T S for every S ∈ L(V ). 46 CHAPTER 4. LINEAR TRANSFORMATIONS 14. Suppose that W is finite dimensional and T ∈ L(V, W ). Prove that T is injective if and only if there exists an S ∈ L(W, V ) such that ST is the identity map on V . 15. Suppose that V is finite dimensional and T ∈ L(V, W ). Prove that T is surjective if and only if there exists an S ∈ L(W, V ) such that T S is the identity map on W .

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