Systems of linear equations

A linear equation has the form $a_1x_1+\cdots+a_nx_n=b$, with constant coefficients (not all zero) and first powers of the chosen unknowns.

The source motivates linear algebra through digital images: pixels form a matrix; its Mona Lisa illustration compares a full image with a 75-by-113 pixel representation, describes 0/1 as black/white and mentions 256 shades.

The original portrait illustration is retained in the source review evidence; the independent grid below illustrates how arrays represent image data. Binary values and a 256-level encoding are different conventions, not simultaneous requirements.

Linearity depends on the chosen unknowns:

Equation	Linear in $x,y,z$?
$2x+y-z=6$	Yes
$10x=5$	Yes
$x+\dfrac{y}{23486}+11.786z=4579$	Yes: all coefficients are constants
$2\ln x-3\ln y+4\ln z=-1$	No; linear in $\ln x,\ln y,\ln z$ instead, with $x,y,z>0$
$\tfrac12\sin x+3\sin y+\sqrt3\sin z=0$	No; linear in $\sin x,\sin y,\sin z$ instead

A system has $m$ equations in $n$ common unknowns. Write row $i$ as $\sum_{j=1}^{n}a_{ij}x_j=b_i$. In matrix form, $A\mathbf{x}=\mathbf{b}$, where $A=(a_{ij})$ is $m\times n$, $\mathbf{x}=(x_1,\ldots,x_n)^T$ and $\mathbf{b}=(b_1,\ldots,b_m)^T$. The augmented matrix $[A\mid\mathbf{b}]$ retains the right-hand column.

The printed example is $x+y+2z=9$, $2x+4y-3z=1$, $3x+6y-5z=0$:

$\begin{pmatrix}1&1&2\\2&4&-3\\3&6&-5\end{pmatrix}\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}9\\1\\0\end{pmatrix},\qquad\left[\begin{array}{ccc|c}1&1&2&9\\2&4&-3&1\\3&6&-5&0\end{array}\right].$