Condition on linear transformations to be non-singular

Question

Let $T$ be a linear transformation over a linear vector space $L$. For $T$ to be non-singular, it has to be one-to-one and onto i.e. a bijection. Since a linear transformation is a homomorphism of the first space onto the second space, does this imply that the $T$ is non-singular only when it is an isomorphism on $L$?

It would be helpful if someone explains in simple mathematical terms, as much as possible, as I am not a student of Mathematics.

Answer 1

An isomorphism is a bijective homomorphism by definition.

Answer 2

Let $T:V\to W$ be a linear transformation. Assuming $V$ and $W$ both to be finite-dimensional, every non-singular linear transformation $T$ is an isomorphism. After all, an isomorphism of vector spaces is a bijective linear transformation. In fact, if dim$V=$dim$W$, then $V\cong W$.

Answer 3

Consider more generally a linear map $T:V\to W.$

1. $T$ is non-singular iff it is one-to-one. This implies $\dim V\le\dim W.$
2. When $\dim V=\dim W<\infty,$ $T$ is one-to-one iff it is onto.
3. When $T$ is bijective, its inverse is linear hence $T$ is an isomorphism.