Are there other Identity Matrices? Is there only one identity matrix $$\begin{pmatrix} 1&0&...&...&0\\0&1&0&...&0\\...&0&1&...&0\\...&...&0&1&0\\...&...&...&0&1\end{pmatrix}$$ etc..
Or are there different identity matrices for other bases? 
A textbook example asks if $[T]_{\beta} = I$ (the $n\times n$ identity matrix) for some basis $\beta$, is $T$ the identity operator? 
 A: Suppose we want
$$
\begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} p & q \\ r & s \end{bmatrix} \begin{bmatrix} a & b \\ c & d \end{bmatrix}
$$
to be true regardless of which matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is, so that $\begin{bmatrix} p & q \\ r & s \end{bmatrix}$ is an identity matrix.  Since it's true regardless of which matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is, it must be true in particular if $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, so we have
$$
\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} =\begin{bmatrix} p & q \\ r & s \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}.
$$
This last equality clearly implies that $\begin{bmatrix} p & q \\ r & s \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$.  Conclusion: if $\begin{bmatrix} p & q \\ r & s \end{bmatrix}$ is an identity matrix, then $\begin{bmatrix} p & q \\ r & s \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$.  Therefore there is only one $2\times2$ identity matrix.  And the same argument works for bigger matrices.
A: By definition, $[Tv]_\beta = [T]_\beta [v]_\beta = [v]_\beta$ for all $v$. Therefore $Tv=v$ for all $v$, i.e. $T$ is the identity on $V$. But to address your real question, here are some more questions for you:


*

*What happens if I try to perform a "change of basis" on the identity matrix?

*What happens if I try to represent the identity operator (on some $V$) in some basis? What does the identity operator do to the basis vectors?

*What is the definition of the identity matrix, and what should this have to do with bases for vector spaces?

