Deciding which properties of linear transformations are true

Question

For a linear transformation $T \colon V \to W$, choose which of the following are correct:

a) $T$ preserves vector space operations.
b) $T$ maps zero vector of $V$ to that of $W$.
c) When $\dim V = m$, $\dim W = n$ and $T$ is invertible then $m$ equals $n$.
d) When $\dim V = m$ and $\dim W = n$ then $T$ is invertible iff $m$ equals $n$.

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I wonder if I'm thinking correctly:

About a), I think it means scalar multiplication and addition, so it's true.
About b), $T$ is linear, so it's true.
About c) and d), if a transformation maps $\mathbb{R}^{n=m}$ to $\mathbb{R}^{n=m}$ then it's invertible, so it's true.

Please correct me if I'm wrong.

Answer 1

If a transformation maps from $\mathbb{R}^m$ to $\mathbb {R}^m$, it need not be invertible. For example, consider the map which takes any element in $\mathbb{R}^m$ to the zero vector. This is neither one-one nor onto.

Answer 2

a),b) and c) are true.

The iff in d) is false. Any map that has nonzero kernel, for instance, won't be invertible. (For a map to be invertible it must be one-one.) Not every map between spaces of the same dimension is invertible. For a trivial example, consider the zero map (kernel is $V$), as @Sudheesh points out... For other examples, consider any map whose rank is less than $n$ (of course, we only need one counterexample)...