The Geometry of Linear Equations

We have a system of equations:

$$\{\begin{array}{rl}2x-y& =0\\ -x+2y& =3\end{array}$$

Row Picture §

Line $2x-y=0$ and line $-x+2y=0$ intersects at the point $(1,2)$, so $(1,2)$ is the solution of the system of equations.

Column Picture §

We rewrite the system of linear equations as a single equation:

$$x\left[\begin{array}{c}2\\ -1\end{array}\right]+y\left[\begin{array}{c}-1\\ 2\end{array}\right]=\left[\begin{array}{c}0\\ 3\end{array}\right]$$

We see $x$ and $y$ as scalars of column vectors: ${\mathbf{v}}_1=\left[\begin{array}{c}2\\ -1\end{array}\right]$ and ${\mathbf{v}}_2=\left[\begin{array}{c}-1\\ 2\end{array}\right]$, and the sum $x{\mathbf{v}}_1+y{\mathbf{v}}_2$ is called a linear combination of ${\mathbf{v}}_1$ and ${\mathbf{v}}_2$.

Geometrically, we can find one copy of ${\mathbf{v}}_1$ added to two copies of ${\mathbf{v}}_2$ just equals the vector $\left[\begin{array}{c}0\\ 3\end{array}\right]$. Then the solution should be $x=1,y=2$.

Matrix Picture §

We rewrite the equations in our example as a compact form,

$$A\mathbf{x}=\mathbf{b},$$

that is

$$\left[\begin{array}{cc}2& -1\\ -1& 2\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}0\\ 3\end{array}\right]$$