Null Space Contains Zero Vector

Theorem
Let:


 * $\operatorname{N}\left({ \mathbf{A} }\right) = \left\{{\mathbf{x} \in \R^n : \mathbf{Ax} = \mathbf 0}\right\}$

be the null space of $\mathbf{A}$, where:


 * $ \mathbf A_{m \times n} = \begin{bmatrix}

a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \\ \end{bmatrix}$

is a matrix in the matrix space $\mathbf M_{m,n}\left({\R}\right)$.

Then the null space of $\mathbf A$ contains the zero vector:


 * $\mathbf 0 \in \operatorname{N}\left({ \mathbf{A} }\right)$

where $\mathbf 0 = \mathbf 0_{m \times 1} = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \end{bmatrix}$.

Proof 1
The dimensions are correct, by hypothesis.

Hence the result, by the definition of null space.

Proof 2
From Matrix Product as Linear Transformation, $\mathbf {Ax} = \mathbf 0$ defines a linear transformation from $\R^m$ to $\R^n$.

The result then follows from Linear Transformation Maps Zero Vector to Zero Vector.

Also see

 * Null Space Closed Under Vector Addition
 * Null Space Closed Under Scalar Multiplication
 * Null Space is Subspace
 * Kernel of Linear Transformation Contains Zero Vector