06. Matrix Equations

Dated: 12-03-2025

A fundamental idea in linear algebra¹ is to view a linear combination² of vectors³ as the product of a matrix⁴ with a vector.³

Definition

If \(A\) is a \(m \times n\) matrix⁴ and \(\vec x\) is a vector³ in \(\mathbb R^n\) then \(A\vec x\) is a linear combination.²

\[ A \vec x = \begin{bmatrix} a_1 & a_2 & \cdots & a_n \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix} = a_1x_1 + a_2x_2 + \cdots + a_nx_n \]

Example

\[ \begin{aligned} x_1 + 2x_2 - x_3 = 4 \\ -5x_2 + 3x_3 = 1 \end{aligned} \]

\[\Big\downarrow\]

\[ x_1 \begin{bmatrix} 1 \\ 0 \end{bmatrix} + x_2 \begin{bmatrix} 2 \\ -5 \end{bmatrix} + x_3 \begin{bmatrix} -1 \\ 3 \end{bmatrix} = \begin{bmatrix} 4 \\ 1 \end{bmatrix} \]

\[\Big\downarrow\]

\[ \begin{bmatrix} 1 & 2 & -1\\ 0 & -5 & 3 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ \end{bmatrix} = \begin{bmatrix} 4 \\ 1 \end{bmatrix} \]

\[\Big\downarrow\]

\[A \vec x = \vec b\]

This is called the matrix equation.

Theorem 1

The solution set for \(A \vec x = \vec b\) is same as the for the equation

\[x_1 \vec a_1 + x_2 \vec a_2 + \cdots + x_n \vec a_n = \vec b\]

Existence of Solution

The equation \(A \vec x = \vec b\) has a solution if and only if \(\vec b\) is a linear combination¹ of the columns of \(A\).

Theorem 2

If \(A\) is a \(m \times n\) matrix⁴ then following statements are logically equivalent.

For each \(\vec b\) in \(\mathbb R^m\) the equations \(A \vec x = \vec b\) has a solution.
The columns of \(A\) span⁵ \(\mathbb R^m\).
\(A\) has a pivot position⁶ in every row.

The Dot Product³

\(\begin{bmatrix} 2 & 3 & 4 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 2x_1 + 3x_2 + 4x_3 \end{bmatrix}\)

06. Matrix Equations

Definition

Example

Theorem 1

Existence of Solution

Theorem 2

References