# Linear Algebra - Null Space of a (Matrix|Vector Space)

### Table of Contents

## 1 - About

Null space of a matrix A (Written Null A) is: <MATH> \{u : A * u = 0\} </MATH>

The Null space of a matrix is a basis for the solution set of a homogeneous linear system that can then be described as a homogeneous matrix equation.

A null space is also relevant to representing the solution set of a general linear system.

As the NULL space is the solution set of the homogeneous linear system, the Null space of a matrix is a vector space.

## 2 - Articles Related

## 3 - Example

- For the Matrix below, the null space includes [0, 0, 0]

<MATH> \begin{bmatrix} 1 & 2 & 4 \\ 2 & 3 & 9 \end{bmatrix} * [0,0,0] = [0,0] </MATH>

- but the null space of this matrix includes also [6,-1,-1]

<MATH> \begin{array}{rrl} \begin{bmatrix} 1 & 2 & 4 \\ 2 & 3 & 9 \end{bmatrix} * [6,-1,-1] & = & [[1, 2, 4] * [6,-1,-1],[2, 3, 9] * [6,-1,-1]] \\ & = & [1*6+ (-1)*2+(-1)*4],2*6 + (-1) *3+ (-1)*9] \\ & = & [0,0] \end{array} </MATH>

## 4 - Null space as a

### 4.1 - solution set of a homogeneous linear system

By matrix-vector dot-product definition (a and u are vectors) <MATH> \begin{bmatrix} \begin{array}{c} a_1 \\ \hline \vdots \\ \hline a_n \\ \end{array} \end{bmatrix} * u = [a_1 * u, \dots, a_m * u] </MATH>

u is in the null space of the matrix if and only if u is a solution to the homogeneous linear system <MATH> \begin{array}{c} a_1 * u = 0 \\ \hline \vdots \\ \hline a_n * u = 0 \end{array} </MATH> As the NULL space is the solution set of the homogeneous linear system, the Null space of a matrix is a vector space.

### 4.2 - vector space

Null space of a matrix is a vector space (see above and example):

<MATH> NULL \begin{bmatrix} 1 & 2 & 4 \\ 2 & 3 & 9 \end{bmatrix} = Span {[6,-1,-1]} </MATH>

### 4.3 - Matrix equation

#### 4.3.1 - Homogeneous

#### 4.3.2 - General

If <math>u_1</math> is a solution to the matrix equation <math>A * x = b</math> then solution set of <math>A * x = b</math> is <math>u_1 + V</math> where <math>V = Null{A}</math>

## 5 - Type

### 5.1 - Trivial

When the rows of A are linearly independent, Null A is trivial and the only solution is <math>u_1</math>.