MATH 108A UCSB Linear Algebra Exam Practice

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Mathematics

MATH 108A

University Of California Santa Barbara

MATH

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Math 108A – Assignment 3 Due: 04/26/21 20h00 PDT Instructions: Label your answers clearly. Create a PDF file, upload it to Gradescope, and instruct Gradescope on which pages to find your answers. In all questions you must provide a proof or justification for your answer. Your work will be graded for mathematical correctness and expository clarity. Please understand that, because of the size of the class and TA workload constraints, only selected problems may be graded. Point values will be determined at the time of grading. (1) Suppose that V and W are vector spaces over the same scalar field F. Prove that if T ∈ L(U, V ) and S ∈ L(V, W ), then ST ∈ L(U, W ). (2) Let V be a vector space over F. Give an example of two linear maps S, T ∈ L(V, V ) such that ST 6= T S. (3) Let V, W be vector spaces over the same scalar field F. Suppose that V is finite dimensional and that {v1 , . . . , vm } is a basis for V . Let T ∈ L(V, W ), and define w1 = T v1 , . . . , wm = T vm . Prove that T is injective iff {w1 , . . . , wm } is linearly independent. (4) Let V and W be vector spaces over the same scalar field F. Let U be a subspace of V . Prove that the set U = {T ∈ L(V, W ) : range T ⊂ U } is a subspace of L(V, W ). (5) Suppose that V is a finite dimensional vector space over F. Let S, T ∈ L(V, V ). Prove that if ST is injective, then S is surjective. (6) Let V be a vector space over F, and let T ∈ L(V, V ). Prove that if T is injective and range T is finite dimensional then V is finite dimensional. (7) Let V = P3 (R) with the standard basis {1, x, x2 , x3 }. Let W = P2 (R) with the basis {x, x + 1, x2 − 1}. Find the matrix of the derivative map D ∈ L(V, W ) relative to these bases. Copyright © 2021 University of California
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