What is the general method of expressing a basis in terms of another basis of a vector space?

Question

I want to understand basis of vector spaces more clearly. I know that the basis of a vector space is a set of linearly independent elements that span that vector space. I know that an element in a vector space can be expressed in different bases. Now most of the examples I have seen give a standard vector $ \left[ \begin {array}{c} b1\\ b2\end {array} \right] $ in terms of a basis $B = [b1,b2]$ that can be expressed in the "standard basis" $S = \{ \left[ \begin {array}{c} 1\\ 0\end {array} \right], \left[ \begin {array}{c} 0\\ 1\end {array} \right] \} $. From here follows that if I have a vector $\left[ \begin {array}{c} 3\\ 5\end {array} \right]$ expressed in terms of $B$ then the change of basis matrix $C$ is the matrix with $b1,b2$ as its columns, therefore in the standard basis $\left[ \begin {array}{c} 3\\ 5\end {array} \right]$ would be expressed as: $$C \left[ \begin {array}{c} 3\\ 5\end {array} \right]$$. Now I want to know that if I have a vector say expressed in a basis $A=[a1,a2]$ how can I write that in terms of the basis $B$? In other words, how does one obtain the change of basis matrix for one basis to another other than the "standard basis"? I have used two dimensional basis, but obviously I am interested in the general n-dimensional case.

Answer 1

Express each element of $B$ as a linear combination of elements of $A$. More precisely, if $B=\{b_1,\ldots,b_n\}$, if $A=\{a_1,\ldots,a_n\}$, and if, for each $i\in\{1,2,\ldots,n\}$,$$b_i=c_{1i}a_1+c_{2i}a_2+\cdots+c_{ni}a_n,$$then$$C=\begin{bmatrix}c_{11}&\ldots&c_{1n}\\\vdots&\ddots&\vdots\\c_{n1}&\ldots&c_{nn}\end{bmatrix}.$$