Question: Let Let (V, ( , )v) and (W, ( , )w) be finite-dimensional inner product spaces. Recall that the adjoint L* : W - V

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Let (V, ( , )v) and (W, ( , )w) be finite-dimensional inner product spaces. Recall that the adjoint L* : W - V of a linear function L E Hom(V, W) is completely determined by the equation (L(v), w)w = (v, L*(w))v for every vEV and we W . Use this to prove the following facts: (a) (L1 + L2) * = Li + 2 for L1, L2 E Hom(V, W) . (b) (aL)* = aL* for a ER and LE Hom(V, W) . (c) (L* ) * = L for LE Hom(V, W) . If (X, ( , ) x) is another finite-dimensional inner product space, and if L E Hom(V, W) and ME Hom(W, X) , then (MOL) * = L* OM *

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