Uniqueness of Linear Transformation from Basis Vectors

[jazz_lover]

Homework Statement

Suppose A is an m x n matrix.

(a) Let v₁ ,...,v_n be a basis of ℝⁿ, and Av_i = w_i ε ℝ^m, for i = 1,...,n. Prove that the vectors v₁,...,v_n, w₁,...,w_n, serve to uniquely specify A.

(b) Write down a formula for A.

Homework Equations

Maybe B = T^-1 A S

The Attempt at a Solution

I said S = { v₁,...,v_n} and T = {w₁,...,w_n}, then AS=T... I kind of get lost from here.

I think part (b) is suppose to be A = TBS^-1... really not too sure though.

 

[CompuChip]

So you can write an arbitrary vector as v = a₁ v₁ + a₂ v₂ + ... + a_n v_n.
Does the given information unambiguously define Av now?
I.e. can you write down the result without reference to A?