Linear Map to Higher-Dimensional Space is Not Surjective

Linear map to a higher-dimensional space is not surjective

Suppose $V$ and $W$ are finite-dimensional vector spaces such that $\dim V<\dim W$. Then, no linear map from $V$ to $W$ is surjectivity.

Let $T\in \mathcal{L}(V,W)$. Then

$$\begin{array}{rl}\dim \text{range }T& =\dim V-\dim \text{null }T\\ & \le \dim V\\ & <\dim W\end{array}$$

where the first equality comes from the fundamental theorem of linear maps. The inequality states that $\dim \text{range }T<\dim W$; this means that $\text{range }T$ cannot equal $W$. Thus $T$ is not surjective.