Hi guys.

Let say \left [ I \right ]_{B}^{C} is the change of basis matrix from the base B to base C.
I need to prove that \left [ I \right ]_{B}^{C} is invertible.

So that's what i did so far:
I defined:
B=\left \{b_1, b_2,... b_n  \right \}
C=\left \{c_1, c_2,... c_n  \right \}

Now I tried to define the \left [ I \right ]_{B}^{C} matrix (and to show its columns are linear independent), but this is where it got a little complicated for me:
since c_i\in V and B is basis for V, every vector c_i\in C can be represented as a linear combination of B's vectors:

c_1=\beta_{11} b_1+\beta_{12} b_2+...+\beta_{1n} b_n
c_2=\beta_{21} b_1+\beta_{22} b_2+...+\beta_{2n} b_n
.
.
.
c_n=\beta_{n1} b_1+\beta_{n2} b_2+...+\beta_{nn} b_n

and then, these equation can be written as:

\begin{bmatrix}c_1\\c_2\\.\\.\\.\\c_n\end{bmatrix}  =\begin{bmatrix}\beta_{11} & \beta_{12} & . & . & . & \beta_{1n}\\\beta_{21} & \beta_{22} & . & . & . & \beta_{2n}\\. & . &  &  &  & .\\. & . &  &  &  & .\\. & . &  &  &  & .\\\beta_{n1} & \beta_{n2} & . & . & . & \beta_{nn}\end{bmatrix}\begin{bmatrix}b_1\\ b_2\\ .\\ .\\ .\\ b_n\end{bmatrix}

Now, where is this change of basis matrix?
Is it the matrix with the \beta_{ij} entries?
Am I even writing it correctly?

Thanks in advanced!