Math 110, Section 7 Fall 2003SarasonREVIEW PROBLEMS1. Let U = {p ∈P(R):p(0)2= p(1)2}.IsU a subspace of P(R)? Explain.2. Let v1,v2,v3,v4beabasis for a vector space V of dimension 4. Prove that v1+ v2+ v3,v1+ v2+ v4, v1+ v3+ v4, v2+ v3+ v4is also a basis for V .3. Let v1,v2,v3be vectors in a vector space V . Assume that span(v1,v2), span(v1,v3) andspan(v2,v3) all have dimension 2. Does it follow that span(v1,v2,v3) has dimension 3? Givea proof or a counterexample.4. Let v1,v2,...,vnbe vectors in a vector space V over F, and let m denote the dimension ofspan(v1,v2,...,vn). Define the subspace U of FnbyU = {(a1,a2,...,an) ∈ Fn: a1v1+ a2v2+ ···+ anvn=0}.Prove that dim U = n − m. (Suggestion: Consider the linear transformation T : Fn→ Vdefined by T (a1,a2,...,an)=a1v1+ a2v2+ ···+ anvn.)5. Let v1,v2,...,vnbe linearly independent vectors in a vector space V , and let w be anothervector in V . Prove that span(v1+ w, v2+ w,...,vn+ w) has dimension at least n − 1.6. Let V beavector space of dimension n>1. Let T1,...,Tnbe linearly independent transforma-tions in L(V ). Is there then a vector v in V such that Tv1,...,Tvnare linearly independent?Give a proof or a counterexample.7. Let V be a finite-dimensional vector space, and let T be in L(V ). Prove that range T =range T2if and only if null T =null T2.8. Let V and W be finite-dimensional vector spaces, and let T be in L(V,W). Assume baseshave 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 ofrange T is smaller than dim V .(c) Prove that the rows of a square matrix are linearly dependent if and only if the columnsare linearly dependent.9. Let V be a finite-dimensional vector space, let U be a subspace of V , and let T be in L(V ).Prove thatdim TU = dim U −dim(U ∩ null T)dim T−1U = dim(null T )+dim(U ∩ range T ).10. Let V beafinite-dimensional vector space and let S be a noninvertible transformation inL(V ). Prove that there is a nonzero transformation T in L(V ) such that ST = TS =0.11. Consider the five-dimensional vector space P4(F), consisting of the polynomials with coef-ficients in F whose degrees are at most 4. Let p0,p1,p2,p3,p4be the standard basis forP4(F):pk(z)=zk(k =0, 1, 2, 3, 4). Let the linear transformation T : P4(F) →P4(F)bedefined by (Tp)(z)=p(z) − p(2z).(a) Find the matrices for T and T2with respect to the standard basis.(b) Is T invertible? Justify your answer.12. Let V be avector space of finite dimension n, and let S beatransformation in L(V ) suchthat null S has dimension m. Determine the dimensions of the subspaces LSand RSof L(V )defined byLS= {T ∈L(V ):TS =0}RS= {T ∈L(V ):ST
View Full Document
Unlocking...