Rank-Nullity Theorem

The Rank-Nullity theorem is a fundamental result in linear algebra that relates the dimension of the kernel and the rank of a matrix.

Statement

Let $A$ be an $m\times n$ matrix. Then, the Rank-Nullity theorem states that:

$$\mathrm{rank}(A)+\dim (\mathrm{Nul}(A))=n$$

where:

• $\mathrm{rank}(A)$ is the dimension of the column space of $A$ (the number of linearly independent columns in $A$).
• $\dim (\mathrm{Nul}(A))$ is the dimension of the null space of $A$ (the number of free variables in the solution to the homogeneous equation $A\vec{x}=\vec{0}$).
• $n$ is the number of columns in the matrix $A$.

Human

This makes initiative sense, as once you bring a matrix to RREF, a column can either be a pivot position or not. If it is a pivot position, it contributes to the rank. If it is not, it contributes to the nullity.

Example

Find a matrix $A$ such that

$$\{\left[\begin{array}{c}b-c\\ 2b+c+d\\ 5c-4d\\ d\end{array}\right],b,c,d\in \mathbb{R}\}$$

We start by setting up the system of equations:

$$\left[\begin{array}{c}b-c\\ 2b+c+d\\ 5c-4d\\ d\end{array}\right]=b\left[\begin{array}{c}1\\ 2\\ 0\\ 0\end{array}\right]+c\left[\begin{array}{c}-1\\ 1\\ 5\\ 0\end{array}\right]+d\left[\begin{array}{c}0\\ 1\\ -4\\ 1\end{array}\right]$$

As $RHS$ represents the solution set as a linear combination of 3 vectors, these vectors can simply be the columns of the desired matrix $A$.

$$A=\left[\begin{array}{ccc}1& -1& 0\\ 2& 1& 1\\ 0& 5& -4\\ 0& 0& 1\end{array}\right]$$