Proving Uniqueness of Invertible Matrix for Linear Spaces with Bases E and F"

[cfnoel]

I'm having issues seeing the method to go with this problem. Here it is:

Suppose that X is a linear space of dimension n, and E = {e1,...,en}, F = {f1,...,fn} are two bases of X. Prove that there is a unique invertible n*n matrix [sij] such that if a vector x belonging to X (I don't know how to make the math symbols or subscripts so please bare with me) has components [ai] with respect to E and components [bj] with respect to F, meaning that

x = Summation( i=1 to n) ai*ei, x = Summation( j=1 to n) bj*fj

then
ai = Summation( j=1 to n) sij*bj.

My book doesn't give examples so I'm having a hard time seeing how to do this problem. If E and F are bases, then they're independent so the components ai and bj are unique. Then sij is just the combination of ai and bj. I'm pretty cloudy and would appreciate all the help I could get. Thanks in advance.

 

[lippyka]

Let [Sij] be the matrix such that ai = $\sum_{j=1}^{n}$ sij*bj.We will now prove that this matrix is unique. Suppose there exists another matrix [Tij] such that ai = $\sum_{j=1}^{n}$ tij*bj. Then, for any i and j, we havesij*bj = ai = $\sum_{j=1}^{n}$ tij*bj $\implies$ sij = $\sum_{j=1}^{n}$ tij*(bj/bj) $\implies$ sij = tij Thus, we have shown that [Sij] = [Tij], which means that the matrix is unique.