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**Unformatted text preview: **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 ∈ Mn,n (R) such that AB = BA for any B ∈ Mn,n (R). 3. Let A ∈ Mn,n (R) be an upper-triangular matrix with ones on the diagonal. Prove that if Ak = I for some k ≥ 1, then A = I. 4. Let f ∈ Mn,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 −1 on the diagonal. 6. Let V be a ﬁnite dimensional vector space and f1 , . . . , fk , g ∈ V ∗ . Prove that
k g ∈ Span {f1 , . . . , fk } ⇐⇒
i=1 N (fi ) ⊂ N (g ). 7. Let P (C) be the vector space (over C) of polynomials with complex coeﬃcients. Let T : P (C) → P (C) be a linear map with the property that T (pq ) = T (p)T (q ) for any pair of polynomials p, q . If T is not injective, prove that its range consists only of constant polynomials. 8. Change C with R everywhere in the previous problem. Is the claim still true? ...

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