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 β is a basis for V . Provethat if T is invertible, then T(β) 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 β = {x2, x, 1} and β0 = {2×2 − x, 3×2 + 1, x2} are ordered bases forP2(R). Find the matrix Q that changes β0coordinates into β coordinates. 6. [5 pts] The set β = 12  −13 is a basis for R2. Let A =2 −14 0 . Find[LA]β. 2. [5 pts) Let F be a field and let A,B e Mnxn(F). Prove that if AB is invertible, then
A 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 that
dim(V) = dim(W). Suppose T :V + W is a linear map and B is a basis for V. Prove
that 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) P(R) and M2x3(R)
(b) V = {A € M2x2(R) : A = A} and R
(©) M2x3(C) and M3x2(C)
5. [3 pts] The sets B = {r?, 0,1} and B’ = {2.rº – 0,3.rº + 1, 2} are ordered bases for
P.(R). Find the matrix Q that changes B’ coordinates into B coordinates.
:{(?)(;’)}
is a basis for R2. Let A =
6. [5 pts) The set B =
[LA].
2-1
4 0
Find