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extraproblems - Math 110 Extra problems GSI Dario 1 Let V W...

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Math 110, Extra problems GSI: Dario 1. Let V, W, Z be finite dimensional vector spaces and f : V W , g : V Z linear maps. Prove that N ( f ) is contained in N ( g ) if and only if there exists a linear map L : W Z such that g = L f . 2. Find all the matrices A M n,n ( R ) such that AB = BA for any B M n,n ( R ). 3. Let A M n,n ( R ) be an upper-triangular matrix with ones on the diagonal. Prove that if A k = I for some k 1, then A = I. 4. Let f M n,n ( F ) * be a linear map such that f ( AB ) = f ( BA ) for any pair of n × n matrices A, B . Prove that f = c · tr for some c F . 5. Let T : V V be a linear map such that T 2 = id (here V is finite dimensional). Prove that there is a basis of V with respect to which T is represented by a diagonal matrix with only 1 and
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