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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.