1. [6 pts] Prove or disprove the following statements.(a) The subset U = {(a1, a2) ∈ R2: a1 = −4a2} is a subspace of R2.(b) The subset W = i ab 0 : a, b ∈ C is a subspace of M2×2(C).

2. [5 pts] Suppose V is a vector space over a field F and let u, v be distinct vectors in V and a be a nonzeroscalar in F. Prove that if {u, v} is a basis for V , then {u, u + av} is a basis for V .

3. [5 pts] Is there a linear map T : P2(R) → R2such that T(1) = (2, 3), T(1 +x) = (−2, 7), and T(1 +x+x2) =(0, 9)? Justify your answer.

4. Suppose U, V , and W are finite dimensional vector spaces over a field F. Let T : U → V and S : V → Wbe linear transformations, and suppose S ◦ T = T0 where T0 is the zero transformation, that is T0(u) = 0Wfor all u ∈ U.(a) [3 pts] Prove R(T) ⊆ N(S).(b) [6 pts] Prove that if T is injective and S is surjective, then dim(W) + dim(U) ≤ dim(V ). (Hint: try touse the Dimension Theorem and part (a) in your proof.)

5. For each of the following subspaces, write down a basis and then state the dimension. For this problem only,you do not have to provide justification for your answer. That is, you do not have to prove your sets arebases.(a) [2 pts] The subspace V of P3(R) defined by V = {a0+a1x+a2x2+a3x3 ∈ P3(R) : a0+a1+a2+a3 = 0}.(b) [2 pts] The subspace W of M3×3(C) consisting of all matrices A such that the diagonal entries in A arezero. That is,W = {A ∈ M3×3(C) : Aij = 0 whenever i = j}.(The subspace W is being considered as a vector space over the field C in this example.)

6. Let V be a vector space over a field F.(a) [3 pts] The statement below is false. Disprove the statement below by providing a counterexample.Make sure to explain why your example is a counterexample.If S is a linearly dependent subset of V , then every vector in S can be written as a linear combinationof other vectors in S.(b) [3 pts] Your classmate thinks this statement is true and that they wrote down a proof. Your counterexample shows it’s false, but they can’t find any errors in their proof below. Find the error and then writea sentence or two explaining why it is an error.“Proof ”. Suppose S is a linearly dependent set. Then by definition of linearly dependent, there existdistinct vectors u1, . . . , un ∈ S and scalars a1, . . . , an ∈ F that are not all zero such thata1u1 + · · · + anun = 0.Since the scalars are not all zero, there exists some index i such that ai 6= 0. Thus we can multiplyboth sides by a−1iand subtract to getui = −a−1ia1u1 − · · · − a−1iai−1ui−1 − a−1iai+1ui+1 − · · · − a−1ianun.Thus ui can be written as a linear combination of other elements in S. Since uiis an arbitrary elementin S, we can conclude every element in S can be written as a combination of other elements in S.