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Vn ) be two bases of E, and let P = (ai j ) be the change of basis matrix from (u1 , . . , un ) to (v1 , . . , vn ), so that n vj = a i j ui , i=1 and let P −1 = (bi j ) be the inverse of P , so that n bj i v j . ui = j=1 Since u∗i (uj ) = δi j and vi∗ (vj ) = δi j , we get n vj∗ (ui ) = vj∗ ( bk i v k ) = b j i , k=1 and thus n vj∗ bj i u∗i , = i=1 and similarly n u∗i ai j vj∗ . = j=1 Since n ϕ∗ = n ϕi u∗i = i=1 we get ϕi vi∗ , i=1 n ai j ϕ i . ϕj = i=1 Comparing with the change of basis n vj = a i j ui , i=1 48 CHAPTER 2.

Un ) and (v1 , . . , vm ) has the property that matrix multiplication corresponds to composition of linear maps. The following proposition states the main properties of the mapping f → M (f ) between L(E; F ) and Mm,n . In short, it is an isomorphism of vector spaces. 34 CHAPTER 2. 12 Given three vector spaces E, F , G, with respective bases (u1 , . . , up ), (v1 , . . , vn ), and (w1 , . . , wm ), the mapping M : L(E; F ) → Mn,p that associates the matrix M (g) to a linear map g: E → F satisfies the following properties for all x ∈ E, all g, h: E → F , and all f : F → G: M (g(x)) = M (g)M (x) M (g + h) = M (g) + M (h) M (λg) = λM (g) M (f ◦ g) = M (f )M (g).

A1 n ) and in the special case where n = 1, we have a column vector , represented as   a1 1  ...  am 1 In these last two cases, we usually omit the constant index 1 (first index in case of a row, second index in case of a column). The set of all m × n-matrices is denoted by Mm,n (K) or Mm,n . An n × n-matrix is called a square matrix of dimension n. The set of all square square matrices of dimension n is denoted by Mn (K), or Mn . 7. MATRICES Remark: As defined, a matrix A = (ai j )1≤i≤m, 1≤j≤n is a family, that is, a function from {1, 2, . 