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601-hw-vii

# 601-hw-vii - MATH 601 AUTUMN 2008 Additional homework VII...

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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 = ( v, w 1 , . . . , v, w s ), 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 , 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 . 4. An endomorphism T : V V is called an involution if T 2 = 1, and a projection if T 2 = T . If dim V < and T is an endomorphism of V , show that T

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