1.1 Linear Equation

If we have an equation: $a\cdot x_1 +b\cdot x_2 +c\cdot x_3 =o$, we can write this in matrix form:

• $\displaystyle \left\lbrack \begin{array}{ccc} a & b & c \end{array}\right\rbrack \cdot \left\lbrack \begin{array}{c} x_1 \\ x_2 \\ x_3 \end{array}\right\rbrack =\left\lbrack \begin{array}{cc} a\cdot x_1 +b\cdot x_2 +c\cdot x_3 \end{array}\right\rbrack =\left\lbrack \begin{array}{c} o \end{array}\right\rbrack$

This is a linear equation, where we have a sequence of variables multiplied by coefficients, more generally, this is a linear equation with $n$ unknown variables, and $n+1$ known coefficients, note the $a$ at the beginning:

• $\displaystyle a+\beta_1 \cdot x_1 +\beta_2 \cdot x_2 +...+\beta_{_{n-1} } \cdot x_{n-1} +\beta_n \cdot x_n =0$

In 2 dimension (with two unknowns), this is a line; in 3 dimension, this is a surface.

1.2 System of Linear Equations

We have a system of linear equations, 3 equations and 3 unknowns:

• $\displaystyle \left\lbrack \begin{array}{ccc} a & b & c\\ d & e & f\\ g & h & i \end{array}\right\rbrack \cdot \left\lbrack \begin{array}{c} x_1 \\ x_2 \\ x_3 \end{array}\right\rbrack =\left\lbrack \begin{array}{cc} a\cdot x_1 +b\cdot x_2 +c\cdot x_3 \\ d\cdot x_1 +e\cdot x_2 +f\cdot x_3 \\ g\cdot x_1 +h\cdot x_2 +i\cdot x_3 \end{array}\right\rbrack =\left\lbrack \begin{array}{c} o\\ p\\ q \end{array}\right\rbrack$

We can define these:

• $\displaystyle W=\left\lbrack \begin{array}{ccc} a & b & c\\ d & e & f\\ g & h & i \end{array}\right\rbrack$

• $\displaystyle X=\left\lbrack \begin{array}{c} x_1 \\ x_2 \\ x_3 \end{array}\right\rbrack$

• $\displaystyle v=\left\lbrack \begin{array}{c} o\\ p\\ q \end{array}\right\rbrack$

1.3 Augmented Form

We can write $W$ and $v$ together like this, this is the augmented matrix of the system of linear equations:

• Augmented Matrix:$\left\lbrack \begin{array}{cccc} a & b & c & |o\\ d & e & f & |p\\ g & h & i & |q \end{array}\right\rbrack$