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By Luther Pfahler Eisenhart

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V = i=1 xn The coordinate map v → [v]B is a linear isomorphism from V onto Fn . 3. If T ∈ L(V ) with matrix MB,B (T ) with respect to B, show that [T v]B = MB,B (T )[v]B , for all v ∈ V . Now again let , be a bilinear form on V , and let A be its matrix with n n respect to the basis B of V . Let v = i=1 xi vi and w = j=1 yj vj be vectors in V . 30), we have n n xi aij yj v, w = i=1 j=1 = x1 ··· = t [v]B A [w]B xn   a11 an1 a1n ··· ann  y  1 ..    . 2 essentially gives us all bilinear forms on V , once we’ve fixed a basis B of V .

Assume that T has characteristic polynomial p(λ) = lj=1 (λ − λj )mj , where λ1 , · · · , λl are the distinct eigenvalues of T . 7, V is the direct sum l Vj , V = j=1 where Vj is the generalized eigenspace ker(T − λj IV )mj . Vj is T -invariant and of course the restriction Nj := (T − λj IV )|Vj = T |Vj − λj IVj is nilpotent. 24)   ..   . 0 Rk where each diagonal block Ri is of the form   0 1 0   ..   .     ..  . 25) Each block Ri above corresponds to the list (Njri vi , . . 3.

Choose any subspace U of V complementary to W , so that V = W ⊕ U . Then define the function G on V by G(w + u) = g(w) for all w ∈ W and all u ∈ U . G is a well-defined linear functional on V such that G|W = g. The restriction map f → f |W is a linear map from V ∗ to W ∗ , and the above shows that it is surjective. Since the map v → fv is a linear bijection from V onto V ∗ , we see that the composition v → fv |W is a surjective linear map from V onto W ∗ . The kernel of this map is clearly W ⊥ .

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