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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.
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