# MATH 115 University of California Los Angeles Linear Algebra & Vector Space Question – Assignment Help

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2. (a) [2 pts] Does the set {(1, 1, 4, 5), (8, 2, 0, 0), (−1, 6, 2, 3)} generate R
4
? Justify

(b) [2 pts] Is the set {1 +x
2
, 1 +x, x−x
2
, 1} linearly independent in P2(R)? Justify

(c) [2 pts] Does the set 1 5
−2 0
,

1 0
0 i

,

0 1

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 of
the 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. Prove
that dim(W1 ∩ W2) ≤ dim(W1).

5. [5 pts] Let U denote the subspace of M3×3(F) consisting of upper triangular 3 × 3
matrices. 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 : R
2 → R
2 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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