1.12. Rank-Nullity Theorem

For a matrix $\mathbf{A}\in {\mathbb{R}}^{m\times n}$ , the rank-nullity theorem states that:

$$\dim (\mathrm{Ker}(\mathbf{A}))+\dim (Range(\mathbf{A}))=\dim ({\mathbb{R}}^n)=n$$

By using the definition of rank and nullity this can be written as follows:

$$\mathrm{Rank}(\mathbf{A})+\mathrm{nullity}(\mathbf{A})=n$$

Interpret this is as follows : There is a balance between rank and nullity of a matrix. In typical scenario we want a matrix to be high rank. Therefore, a higher rank matrix will have a low-dimensional null space. The converse is, a low rank rank will have a high dimensional null space.

Visually it can be represented as follows: