I'm looking for nice applications of Rank–nullity theorem to show to my students for a 45 minutes class. I'm going to begin this class showing the demonstration of the theorem and I don't know what to do in rest of the time, any suggestions?
Let $V$ and $W$ be finite-dimensional vector spaces and let $T:V\to W$ be a linear transformation from $V$ into $W$. Then:
$$\dim V=\dim \text{Im}(T)+\dim\ker(T)$$
Remark: The only nice applications I remember is the fact the row and column ranks of a matrix are equal and a linear transformation $T:V\to W$ with $\dim V=\dim W$ is injective iff it's surjective.
Best Answer
A bit too long for a comment ...
Your memory is faulty. It’s easy to prove that $T$ is injective iff $\ker(T)=\{0\}$.
Now the theorem reveals that if $\dim(V)>\dim(W)$ the map $T$ can’t be injective since the dimension of the image of $T$ is at most the dimension of $W$.
OTOH if $\dim(V)<\dim(W)$ then $T$ can’t be surjective for the same reason.
Now furthermore if the dimensions are equal your statement is right.