I am working on the proof of a map to a smaller dimensional space is not injective.
Proof:
Let $T \in \mathcal{L}(V,W)$. Then
$$\text {dim null } T = \text {dim } V – \text {dim range } T \geq \text {dim } V – \text {dim } W \gt 0$$
I don't understand why $\text {dim } V – \text {dim range } T \geq \text {dim } V – \text {dim } W$. Or why $\ \text {dim range } T \leq \text {dim } W$?
I guess it is because as range $T$ is coming from $V$, at most you can get the dimension of $V$. As dim $V$ is less than dim $W$, dim range $T$ is also less than dim $W$. I don't know if it is correct, and some better way to look at it.
Best Answer
By definition, $\text{range} (T)$ is a subspace of $W$. $$\text{range} (T) \le W$$ So, $$\text{dim} \,\text{range} (T) \le \text{dim} W$$