1) Let A be an m × n matrix. Find all b such that the set of solutions to Ax = b is a subspace and prove your claim.

1) Let A be an m × n matrix. Find all b such that the set of solutions to Ax = b is a subspace and prove your claim.

2) Prove or disprove: If T is 1-1, then T^2 is 1-1.

3) Prove or disprove: If T^2 is 1-1, then T is 1-1.

4) Prove or disprove: R^2 with coordinate-wise addition and scaling defined as c(x1, x2) = (x1, 0) is a vector space.

5) Suppose {v1, ..., vn} is a basis for V . Find conditions on c1, ..., cn so that {c1v1, ..., cnvn} is a basis for V and prove your claim.