(6 points) Prove that B={x^(n)|ninN}, the set of all powers of x, is a linearly independent subset of F un (R). Note: B is an infinite set.
(28 points) In class, we proved that if A|ar (a) | and B(||)/(b)ar (b) are row equivalent m imes (n+1) matrices, then they have the same solution set. In this problem, we will prove the converse. The proof combines several ideas from the course. For the duration of this problem, let A|ar (a) | and B(||)/(b)ar (b) be consistent m imes (n+1) matrices. Assume they have the same solution set S.
(a) (6 points) Let V be a vector space over a field F. Let A be an affine subspace of V. By definition, A is a translate of some subspace W of V. Prove this subspace is unique, i.e. if A is also a translate of a subspace W^('), that W=W^(').
subset of FunR).Note:B is an infinite set.
6. (28 points) In class, we proved that if [A|a] and [B|0] are row equivalent m (n + 1) matrices, combines several ideas from the course. For the duration of this problem, let [A|a] and [B|b] be consistent m (n + 1) matrices. Assume they have the same solution set S. (a)(6 points) Let V be a vector space over a field F.Let A be an affine subspace of V.By definition,A is a translate of some subspace W of V.Prove this subspace is unique, i.e. if A is also a translate of a subspace W', that W -W'.
