Solution Found!
Let V and W be finite-dimensional vector spaces with ordered bases 0 = {v i> v2, - - -
Chapter 2, Problem 21(choose chapter or problem)
Let V and W be finite-dimensional vector spaces with ordered bases 0 = {v i> v2, - - -, Vn} and 7 = {w\, w2, - - -, wm}, respectively. By Theorem 2.6 (p. 72), there exist linear transformations T,j: V > W such that Tij(vk) = Wi if k = j 0 iik^ j. First prove that {Tij: 1 < i < m, 1 < j < n} is a basis for (V, W). Then let MlJ be the mxn matrix with 1 in the ith row and jth column and 0 elsewhere, and prove that [T^-H = MlK Again by Theorem 2.6, there exists a linear transformation $ : (V, W) > MmXn (F) such that $(Tij) = My '. Prove that $ is an isomorphism.
Questions & Answers
QUESTION:
Let V and W be finite-dimensional vector spaces with ordered bases 0 = {v i> v2, - - -, Vn} and 7 = {w\, w2, - - -, wm}, respectively. By Theorem 2.6 (p. 72), there exist linear transformations T,j: V > W such that Tij(vk) = Wi if k = j 0 iik^ j. First prove that {Tij: 1 < i < m, 1 < j < n} is a basis for (V, W). Then let MlJ be the mxn matrix with 1 in the ith row and jth column and 0 elsewhere, and prove that [T^-H = MlK Again by Theorem 2.6, there exists a linear transformation $ : (V, W) > MmXn (F) such that $(Tij) = My '. Prove that $ is an isomorphism.
ANSWER:Step 1 of 4
Using the transformation linearity 
Therefore, one can say that 