Math 110 Section 7 Sarason Fall 2003 REVIEW PROBLEMS 1 Let U p P R p 0 2 p 1 2 Is U a subspace of P R Explain 2 Let v1 v2 v3 v4 be a basis for a vector space V of dimension 4 Prove that v1 v2 v3 v1 v2 v4 v1 v3 v4 v2 v3 v4 is also a basis for V 3 Let v1 v2 v3 be vectors in a vector space V Assume that span v1 v2 span v1 v3 and span v2 v3 all have dimension 2 Does it follow that span v1 v2 v3 has dimension 3 Give a proof or a counterexample 4 Let v1 v2 vn be vectors in a vector space V over F and let m denote the dimension of span v1 v2 vn De ne the subspace U of Fn by U a1 a2 an Fn a1 v1 a2 v2 an vn 0 Prove that dim U n m Suggestion Consider the linear transformation T Fn V de ned by T a1 a2 an a1 v1 a2 v2 an vn 5 Let v1 v2 vn be linearly independent vectors in a vector space V and let w be another vector in V Prove that span v1 w v2 w vn w has dimension at least n 1 6 Let V be a vector space of dimension n 1 Let T1 Tn be linearly independent transformations in L V Is there then a vector v in V such that T v1 T vn are linearly independent Give a proof or a counterexample 7 Let V be a nite dimensional vector space and let T be in L V Prove that range T range T 2 if and only if null T null T 2 8 Let V and W be nite dimensional vector spaces and let T be in L V W Assume bases have been chosen for V and W a Prove that the rows of M T are linearly dependent if and only if null T is nontrivial b Prove that the columns of M T are linearly dependent if and only if the dimension of range T is smaller than dim V c Prove that the rows of a square matrix are linearly dependent if and only if the columns are linearly dependent 9 Let V be a nite dimensional vector space let U be a subspace of V and let T be in L V Prove that dim T U dim U dim U null T dim T 1 U dim null T dim U range T 10 Let V be a nite dimensional vector space and let S be a noninvertible transformation in L V Prove that there is a nonzero transformation T in L V such that ST T S 0 11 Consider the ve dimensional vector space P4 F consisting of the polynomials with coef cients in F whose degrees are at most 4 Let p0 p1 p2 p3 p4 be the standard basis for P4 F pk z z k k 0 1 2 3 4 Let the linear transformation T P4 F P4 F be de ned by T p z p z p 2z a Find the matrices for T and T 2 with respect to the standard basis b Is T invertible Justify your answer 12 Let V be a vector space of nite dimension n and let S be a transformation in L V such that null S has dimension m Determine the dimensions of the subspaces LS and RS of L V de ned by LS T L V T S 0 RS T L V ST 0
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