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

About

Null space of a matrix A (Written Null A) is: $$\{u : A * u = 0\}$$

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.

Example

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

$$\begin{bmatrix} 1 & 2 & 4 \\ 2 & 3 & 9 \end{bmatrix} * [0,0,0] = [0,0]$$

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

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

Null space as a

solution set of a homogeneous linear system

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

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

vector space

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

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

Matrix equation

General

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

Type

Trivial

When the rows of A are linearly independent, Null A is Linear Algebra - Linear combination and the only solution is $u_1$ .

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