Null Space

Definition

The Null space of an $m-by-n$ matrix $A$ is the collection of those vectors in ${\mathbb{R}}^n$ that $A$ maps to the zero vector in ${\mathbb{R}}^m$. More precisely:

$$\mathcal{N}(A)=\{x\in {\mathbb{R}}^n\mid Ax=0\}$$

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As an example, let’s examine the matrix $A$:

$$A=\left(\begin{array}{cccc}0& 1& 0& 0\\ -1& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)$$

From this we can find that the null space of the matrix is the following:

$$\mathcal{N}(A)=\{t\left(\begin{array}{c}1\\ 0\\ 1\\ 0\end{array}\right)\mid t\in \mathbb{R}\}$$

This is a line in ${\mathbb{R}}^4$

The null space answers the question of uniqueness of solutions to $Sx=f$. For, if $Sx=f$ and $Sy=f$ then $S(x-y)=Sx-Sy=f-f=0$ and so $(x-y)\in \mathcal{N}(S)$. Hence, a solution to $Sx=f$ will be unique if ,and only if, $\mathcal{N}S=\{0\}$