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Unformatted text preview: MATH 601, AUTUMN 2008 Additional homework VII, November 19 PROBLEMS 1. Let V be an inner-product space of dimension n , and let v be a nonzero vector in V . Use the rank-nullity theorem to show that dim v ⊥ = n- 1. 2. Let W be a subspace of an inner-product space V of dimension n , and let w 1 , . . . , w s be a basis of W . For the linear operator T : V → K s given by Tv = ( h v, w 1 i , . . . , h v, w s i ), show that T restricted to W is an isomorphism W → K s . Conclude from this that T : V → K s is surjective, and then use the rank-nullity theorem to obtain a proof (different from the one shown in class) that dim W ⊥ = dim V- dim W . 3. Using the standard (sesquilinear) Hermitian inner product h , i on C 4 , and letting V ⊂ C 4 be the subspace consisting of all ( x, y, z, t ) , with x + z- it = x- iy- z = 0, write an explicit formula for the V and V ⊥ components (relative to the decomposition C 4 = V ⊕ V ⊥ ) of each ( x, y, z, t ) ∈ C 4 ....
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