Let b = {x_1,...,x_n} be a basis for a vector space V over a field F.(adsbygoogle = window.adsbygoogle || []).push({});

let x'_j = Sum i=1 to n[Bij x_j] where Bij are the entries of any matrix (n x n). Prove that b'={x'_1,...,x'_n} is a basis for V and therefore B is the change of coordinate matrix.

Ok, the final connection between the proof and the coordinate matrix is obvious (that's how its defined). But im not sure how to prove that b' is a basis. I know that

Sum j=1 to n[a_j x'_j]

= Sum j=1 to n[a_j Sum i=1 to n[Bij x_i]]

= Sum j=1 to n[Sum i=1 to n[a_j Bij] x_i]

So you can consider the Sum i=1 to n[a_j Bij] as the coefficient, call it c_i such that each element of the vector space v = Sum i=1 to n[c_i x_i] for unique scalars c_1,...,c_n.

But this would mean b' is a basis only if for each combination c_1,...,c_n there exists a a_1,...,a_n that generates this c_1,...,c_n. Im having trouble showing this.

I know theres another way to look at it. For example, what i just said is equivalent to having:

c_1 = Sum j=1 to n[a_j B1j]

c_2 = Sum j=1 to n[a_j B2j]

.

.

.

c_n = Sum j=1 to n[a_j Bnj]

so if i can prove that given these n equations and n unknows, there exists for every combination, c_1,...c_n a solution for j_1,...,j_n then i would be done.

But the book im reading still didnt get to the chapter about solving equations so there must be a more elementary way of showing this.

So ya, if anyone could help that would be cool.

edit: the matrix is invertible

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# Homework Help: Linear algebra

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