The change of basis matrix comes into play when we wish to write vectors in terms of a new basis.

Assume you some vector $v$ from a vector space which has a basis $B_1 = {b_1, b_2, \dots, b_k}$.

So, you can write your vector $v$ as a linear combination of the basis with some scalars:

$v = \alpha_1b_1 + a_2b_2 + \dots + \alpha_kb_k$

Say you have another basis for the space, $B_2$. You can also write your vector in terms of the other basis, but you want a clear way to do this. Enter the change of basis matrix, which is derived by looking at how we can rewrite elements of one basis as a linear combination of elements from the other basis.

The power of the matrix is that we can ignore the vector you wish to rewrite and instead derive a matrix that will work on vectors from your space.

(To be continued)

Definitions

Orthonormal Basis:

The orthonormal basis definition : it is a set of linearly independent vectors which span the space in question: $e_i \cdot e_i = 1$ and $e_i \cdot e_j$ = 0.

thus, the standard basis in $\mathbb{R^3}$ is an orthonormal basis (0s everywhere except for a 1 in the ith coordinate place)