Suppose $T: V \to V$ is the identity transformation.
If $B$ is a basis of V, then the matrix representation of $[T]^B_B = [I_n]$.
Let's say C is also a basis of V, then it is clear that $[T]^B_C \ne [I_n]$.
However, I was taught that matrices representing the same linear transformation in different bases are similar, and the only matrix similar to $I_n$ is $I_n$. Thus, $[T]^B_C$ and $[T]^B_B$ are not similar.
Can anyone clear what seems to be a contradiction?
Best Answer
What the statement “matrices representing the same linear transformation in different bases are similar” means is that if $B$ and $C$ are bases, then $[T]_B^B$ and $[T]_C^C$ are similar.