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L(A)n (un ) = an 1 u1 + · · · + an n un , for all u1 , . . , un ∈ E. It is immediately verified that L(A) is linear. 12), we can show that L(AB) = L(A) ◦ L(B). It is then convenient to use the matrix notation to describe the effect of the linear map L(A), as    u  L(A)1 (u1 ) 1 a1 1 a1 2 . . a 1 n    L(A)2 (u2 )   a  u2    2 1 a2 2 . . a 2 n    . =    . . ... ..   ..  ..    ..  . an 1 an 2 . . 2. 4 Let f : E × . . × E → F be an n-linear alternating map. Let (u1 , .

24 shows that the dimension of any subspace V such that E = U ⊕ V depends only on U . We call dim(V ) the codimension of U , and we denote it by codim(U ). A subspace U of codimension 1 is called a hyperplane. The notion of rank of a linear map or of a matrix is an important one, both theoretically and practically, since it is the key to the solvabilty of linear equations. 19 Given two vector spaces E and F and a linear map f : E → F , the rank rk(f ) of f is the dimension dim(Im f ) of the image subspace Im f of F .

THE DUAL SPACE E ∗ AND LINEAR FORMS 49 (a) The linear map cE : E → E ∗∗ defined such that cE (v) = v˜, that is, cE (v)(u∗ ) = u∗ , v for every u∗ ∈ E ∗ , is injective. (b) When E is of finite dimension n, the linear map cE : E → E ∗∗ is an isomorphism (called the canonical isomorphism). Proof . (a) Let (ui )i∈I be a basis of E, and let v = cE (v)(u∗i ) = 0 for all u∗i , and since i∈I vi ui . If cE (v) = 0, then in particular, cE (v)(u∗i ) = u∗i , v = vi , we have vi = 0 for all i ∈ I, that is, v = 0, showing that cE : E → E ∗∗ is injective.

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Basics of Algebra and Analysis for Computer Science by Gallier J.


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