09 - Rank-Nullity Theorem

Rank-Nullity Theorem

If $T:V\to W$ is linear transformation and $V$ is finite dimensional then

$$\dim (V)=\dim (\mathrm{Ker}(T))+\dim (\mathrm{Im}T)$$

Proof

Apply dimension formula for quotient spaces to $U=\ker T$ then

$$\dim (V)=\dim (\mathrm{Ker}(T))+\dim (V/\mathrm{Ker}(T))$$

But by First Isomorphism Theorem - V2 then

$$\dim (V/\mathrm{Ker}(T))=\dim (\mathrm{Im}(T)$$

So the result follows