Let $T:V\to W$ on finite dimensional vector spaces, is it possible to use the determinant to determine that $T$ is invertible.

Question

Let $T: V\to W$ be a linear transformation on finite dimensional vector spaces. Then is it possible to use the determinant to determine that $T$ is invertible?

For instance I have $T:\mathbb{C}^3\to \mathbb{C}^3$ $$Te_1=(1,0,i)$$ $$Te_2=(0,1,1)$$ $$Te_3=(i,1,0)$$ then if I make the matrix of transformation $[T]_{DB}$ relative to the bases $B=\{e_1,e_2,e_3\}$ and $D=\{(1,0,i),(0,1,1),(i,1,0)\}$ assuming $D$ forms a basis then if I show that $\det [T]_{DB}\not =0$ will that imply that $T$ is invertible?

Answer 1

Yes, of course. If the determinant of $[T]_{DB}$ is not $0$, then the determinant of $[T]_{BB}$ is also not $0$, and therefore $T$ is invertible. Furthermore, $[T^{-1}]_{BB}=\left([T]_{BB}\right)^{-1}$.