When can we say that a linear transformation is equivalent to a change of basis?

Question

I'm aware that every change of basis is a linear transformation, but the converse isn't true. What conditions must a linear transformation $T$ satisfy for us to call it a change of basis? One condition that I can think of is that $T$ should be invertible, but I'm not sure that's enough to call it a change of basis.

Answer 1

Invertibility is equivalent to the span's dimension staying the same, which in turn is equivalent to the linear transformation giving a basis.

Let the original basis $B$ have $n$-dimensional span. (This is just the vector space's dimension. If the elements of $B$ are linearly independent so that $B$ has a minimal number of elements, $B$ has $n$ elements.) The dimension of $\operatorname{span}\{Tb|b\in B\}$ is then the rank of $T$, which is $n$ iff $T$ is invertible.

Answer 2

Any invertible matrix can be interpreted as a change of basis between the standard basis and the basis consisting of its columns. To effect the change of basis, conjugate by the matrix.

So, if $P$ is said matrix, $P^{-1}AP$ will be the matrix $A$ in the new basis.

Since $P$ transforms from the basis consisting of its columns to the standard basis, vectors are described as contravariant.

Whereas the conjugation described above on matrices is $1$-co-, $1$-contra-variant.