MATH 115 University of California Los Angeles Linear Algebra & Vector Space Question
Question Description
2. (a) [2 pts] Does the set {(1, 1, 4, 5), (8, 2, 0, 0), (−1, 6, 2, 3)} generate R4? Justifyyour answer.
(b) [2 pts] Is the set {1 +x2, 1 +x, x−x2, 1} linearly independent in P2(R)? Justifyyour answer.
(c) [2 pts] Does the set 1 5−2 0 , 1 00 i , 0 1−2 0 span the complex vector space M2×2(C)? Justify your answer.
3. [6 pts] Let V be a vector space and let v1, . . . , vk, v ∈ V . Define W1 = span{v1, . . . , vk}and W2 = span{v1, . . . , vk, v}. Prove that v can be written as a linear combination ofthe vectors v1, . . . , vk if and only if dim(W1) = dim(W2).
4. [4 pts] Let V be a vector space with finite dimensional subspaces W1 and W2. Provethat dim(W1 ∩ W2) ≤ dim(W1).
5. [5 pts] Let U denote the subspace of M3×3(F) consisting of upper triangular 3 × 3matrices. Find a basis for U (make sure to prove your choice is actually a basis).What is dim(U)?
6. [4 pts] Suppose T is a function T : R2 → R2 with T((2, 6)) = (2, −2) and T((1, 3)) =(1, −5). Is it possible that T is a linear map? Why or why not?
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