Linear and multilinear algebra; matrix theory Proposition

Matrix equation, linear combination of columns and linear system

Builds on Matrix–vector product Linear combination Augmented matrix System of linear equations

Statement

Let $A$ be an $m \times n$ matrix with columns $v_1, \dots, v_n$ . Let $x = (x_1, \dots, x_n)^T$ be a column vector with $n$ entries, and let $b$ be a column vector with $m$ entries.

Then the following three statements are equivalent.

The vector $x$ satisfies the equation $Ax = b$ , using the matrix-vector product.
The vector $b$ is the linear combination $x_1 v_1 + \dots + x_n v_n$ of the columns of $A$ , with the entries of $x$ as weights.
The entries $x_1, \dots, x_n$ form a solution of the linear system whose augmented matrix is $[v_1 \; \dots \; v_n \mid b]$ .

Remarks

The product $Ax$ is defined as the combination of the columns of $A$ weighted by the entries of $x$ . This gives the link between the first two statements. The third statement reads the same equation one row at a time.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Augmented matrix concept
Linear combination concept
Matrix–vector product concept
System of linear equations concept
Matrix equation, linear combination of columns and linear system statement

Tag	`E7SAHXQEVC`
URI	`m2://E7SAHXQEVC`
Kind	statement, proposition
Writing	W0 stub
MSC	15A06
Status	This copy of the site has no build data. Run `m2 status E7SAHXQEVC`.
Uses	Matrix–vector product, Linear combination, Augmented matrix, System of linear equations
Changed	4 November 2026

Matrix equation, linear combination of columns and linear system

Statement

Remarks

Reading path

Further reading

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