# Linear Algebra - Matrix Equation

### Table of Contents

## 1 - About

A linear vector equation is equivalent to a matrix equation of the form : <MATH>A.x=b</MATH>

where:

- A is an m×n matrix,
- x is a column vector with n entries,
- and b is a column vector with m entries.

<MATH> A= \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},\quad \bf{x}= \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix},\quad \bf{b}= \begin{bmatrix} b_1 \\ b_2 \\ \vdots \\ b_m \end{bmatrix} </MATH>

A matrix equation is the same:

- as expressing a given vector (b) as a linear combination of other given vectors (A*x),
- and therefore as solving a system of linear equations

## 2 - Articles Related

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## 3 - Type

### 3.1 - Homogeneous

A set of a homogeneous linear system as a matrix equation <MATH> Ax = 0 </MATH>

This formulation describes the Null Space of A

## 4 - Computation

Python Scipy can solve a linear system Ax = b with linalg.solve(A, b)