UCLA Isomorphic Dimensional Vector Spaces and Matrix Coordinates Exercises

2. [5 pts] Let F be a field and let A, B ? Mn×n(F). Prove that if AB is invertible, thenA is invertible and B is invertible. (Hint: Consider the transformations LA and LB.)

3. [5 pts] Let V and W be finite dimensional vector spaces over a field F such thatdim(V ) = dim(W). Suppose T : V ? W is a linear map and ? is a basis for V . Provethat if T is invertible, then T(?) is a basis for W.

4. [7 pts] Determine whether or not the following pairs vector spaces are isomorphic.Write a sentence or two to justify your answer.(a) P5(R) and M2×3(R)(b) V = {A ? M2×2(R) : At = A} and R4(c) M2×3(C) and M3×2(C)

5. [3 pts] The sets ? = {x2, x, 1} and ?0 = {2×2 ? x, 3×2 + 1, x2} are ordered bases forP2(R). Find the matrix Q that changes ?0coordinates into ? coordinates.

6. [5 pts] The set ? = 12 ?13 is a basis for R2. Let A = 2 ?14 0 . Find[LA]?.