Question

Problem 5. A subset A C R is an affine subspace of R" if there exists a vector bE R" and an underlying vector subspace W of R" such that (a) Describe all the affine subspaces of R2 which are not vector subspaces of R2 (b) Consider A E Rnx, bER" and the system of linear equations AT . Prove that: (i) if Ais consistent, then its solution set is an affine subspace of R" with underlying (ii) if At - b is inconsistent, then its set of least-square solutions is an affine subspace of R" (c) Find an mx n matrix A and a vector b є Rn such that the linear system Ar-bis inconsistent vector subspace ker(A). (Hint: consider the difference of two solutions). Describe the underlying vector space of this affine subspace. and is the set of least-squares solutions to Ai-b. (No justification needed). Are the A and b you found unique?

Answer 1

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