Rank-nullity theorem

The rank-nullity theorem is one of the most important theorems in linear algebra. It links the fundamental subspaces, the image and kernel together.

For some linear transformation $T_A:{\mathbb{R}}^n\to {\mathbb{R}}^m$:

$$\text{dim(im}(T_A)+\text{dim(ker}(T_A)=n$$ $$\text{rank}(A)+\text{nullity}(A)=n$$

So the rank of a matrix (which we saw in Gaussian elimination) is the dimension of the image of a transformation. And the dimension of the nullspace is called the nullity.