By S. E. Payne

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**Example text**

Let S ∈ L(U, V ) and T ∈ L(V, W ), so that T ◦ S ∈ L(U, W ), where T ◦ S means do S first. Then we have [(T ◦ S)(u)]B3 = [T (S(u))]B3 = [T ]B3 ,B2 [S(u)]B2 = [T ]B3 ,B2 [S]B2 ,B1 [u]B1 = = [T ◦ S]B3 ,B1 [u]B1 for all u ∈ U. This implies that [T ◦ S]B3 ,B1 = [T ]B3 ,B2 · [S]B2 ,B1 . This is the equation that suggests that the subscript on the matrix representing a linear map should have the basis for the range space listed first. Recall that L(U, V ) is naturally a vector space over F with the usual addition of linear maps and scalar multiplication of linear maps.

When U = V and B1 = B2 it is sometimes the case that we write [T ]B1 in place of [T ]B1 ,B1 . And we usually write L(V ) in place of L(V, V ), and T ∈ L(V ) is called a linear operator on V . 6. MATRICES AS LINEAR TRANSFORMATIONS 39 space so that it has a coordinate matrix with respect to some basis of that vector space. Our convention makes it easy to recognize when the matrix represents T with respect to a basis as a linear map and when it represents T as a vector itself which is a linear combination of the elements of some basis.

N . This requires k = j − i interchanges of adjacent rows. We now move αj to the ith position using (k − 1) interchanges of adjacent rows. We have thus obtained B from A by 2k − 1 interchanges of adjacent rows. 4, D(B) = −D(A). Suppose A is any n × n matrix with two equal rows, say αi = αj with i < j. If j = i + 1, then A has two equal and adjacent rows, so D(A) = 0.