MAT A22 Winter 2024 Problems Q1. Let S V be a subset of a vector space. Recall that the span of S is: span(S) = {a1vy + asve + - -+ + ap v, : a; R and v; S} Alternatively, span(.S) is the set of finite linear combinations of el- ements from S. Prove the following: (a) S C span(S) (b) If S C T then span(S) C span(T) (c) span(S) = span(span(5)) Q2. Prove the following;: (a) If S C T and S is linearly dependent then T is linearly depen- dent. (b) If S T and T is linearly independent then S is linearly independent. Q3. A subset S' of S is proper if S' is not equal to S. We write 8" C to indicate that S' is a proper subset of S. Prove: If S is linearly independent and W = span(S) then no proper subset S' of S spans W. Q4. Suppose S = {vy, vy, ..., v, } is linear independent in V. Prove that T = {vy,vi + vo,vi + Vo +V3,..., Vi + -+- + Vv, } is linearly independent in V. Q5. Let W C V be a subset of a vector space. Prove: W is a subspace if and only if W = span(W). Page 3 MAT A22 Winter 2024 Q6. (***This is the question that you must complete using BTRX***) Prove span({1,1+ z,1 + z + 2?}) = P(R) Q7. Suppose {u,v,w} is linearly independent in V. Prove that {u+v,u+w,v+w} is also linearly independent in V