Injectivity, Surjectivity and Isomorphism in Finite Dimensions

In the setting of Linear Algebra: Standing Notation, let $K$ be a field and let $V$ and $W$ be finite-dimensional vector spaces over $K$.

(i) Injective maps. If $T$ is an injective linear map from $V$ to $W$, then $\dim V\le\dim W$.

(ii) Surjective maps. If $T$ is a surjective linear map from $V$ to $W$, then $\dim V\ge\dim W$.

(iii) Equal dimensions. If $\dim V=\dim W$ and $T$ is a linear map from $V$ to $W$, then

$$T\text{ is injective}\iff T\text{ is surjective}\iff T\text{ is bijective}.$$

(iv) Isomorphic spaces. $V$ and $W$ are isomorphic if and only if $\dim V=\dim W$.

(v) Operators. For every $T\in\mathcal{L}(V)$, with units taken in the monoid $\mathcal{L}(V)$ of Linear Maps Form a Vector Space and a Monoid under Composition (iv),

$$T\text{ is a unit of }\mathcal{L}(V)\iff\ker T=\{0\}\iff\operatorname{im}T=V.$$