Linear and multilinear algebra; matrix theory Definition

Coefficient matrix

Builds on System of linear equations Matrix

Definition

Let $F$ be a field. Take a system of linear equations over $F$ with $m$ equations in $n$ unknowns, where equation $i$ is $a_{i1} x_1 + \cdots + a_{in} x_n = b_i$ . The coefficient matrix of the system is the $m \times n$ matrix $A = (a_{ij})$ . Its entry in row $i$ and column $j$ is the coefficient of $x_j$ in equation $i$ .

Remarks

The system $2x + y = 5$ , $x - 3y = 0$ has the coefficient matrix $\begin{pmatrix} 2 & 1 \\ 1 & -3 \end{pmatrix}$ . With $x = (x_1, \dots, x_n)^T$ and $b = (b_1, \dots, b_m)^T$ , the system is $Ax = b$ . If $A$ is square and invertible, the system has exactly one solution, $x = A^{-1} b$ .

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Linear equation definition
Matrix definition
System of linear equations definition
Coefficient matrix definition

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Tag	`0P7A2MJD24`
URI	`m2://0P7A2MJD24`
Kind	definition
Writing	W0 stub
MSC	15A06
Defines	Coefficient matrix
Uses	System of linear equations, Matrix

Coefficient matrix

Definition

Remarks

Reading path

Where this leads

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