40.1: Linear Systems

In this course, we learned how to represent linear systems which basically consists of equations added sums of multiple numbers in the form:

$$b = a_1x_1+a_2x_2+a_3x_3 + \ldots a_mx_m \nonumber$$

Systems of linear equations are multiple equations of the above form with basically the same unknowns but different values of $a$ and $b$.

$$b_1 = a_{11}x_1+a_{12}x_2+a_{13}x_3 + \ldots a_{1n}x_n \nonumber$$

$$b_2 = a_{21}x_1+a_{22}x_2+a_{23}x_3 + \ldots a_{2n}x_n \nonumber$$

$$b_3 = a_{31}x_1+a_{32}x_2+a_{33}x_3 + \ldots a_{3n}x_n \nonumber$$

$$\vdots \nonumber$$

$$b_m = a_{m1}x_1+a_{m2}x_2+a_{m3}x_3 + \ldots a_{mn}x_n \nonumber$$

The above equations can be represented in matrix form as follows:

$$\begin{split} \left[ \begin{matrix} b_1 \\ b_2 \\ b_3 \\ \vdots \\ b_m \end{matrix} \right] = \left[ \begin{matrix} a_{11} & a_{12} & a_{13} & & a_{1n} \\ a_{21} & a_{22} & a_{23} & \ldots & a_{2n} \\ a_{31} & a_{32} & a_{33} & & a_{3n} \\ & \vdots & & \ddots & \vdots \\ a_{m1} & a_{m2} & a_{m3} & & a_{mn} \end{matrix} \right] \left[ \begin{matrix} x_1 \\ x_2 \\ x_3 \\ \vdots \\ x_n \end{matrix} \right] \end{split} \nonumber$$

Which can also be represented in “augmented” form as follows:

$$\begin{split} \left[ \begin{matrix} a_{11} & a_{12} & a_{13} & & a_{1n} \\ a_{21} & a_{22} & a_{23} & \ldots & a_{2n} \\ a_{31} & a_{32} & a_{33} & & a_{3n} \\ & \vdots & & \ddots & \vdots \\ a_{m1} & a_{m2} & a_{m3} & & a_{mn} \end{matrix} \, \middle\vert \, \begin{matrix} b_1 \\ b_2 \\ b_3 \\ \vdots \\ b_m \end{matrix} \right] \end{split} \nonumber$$

The above systems can be modified into equivelent systems using combinations of the following operators.

1. Multiply any row of a matrix by a constant
2. Add the contents of one row by another row.
3. Swap any two rows.

Often the 1st and 2nd operator can be combined where a row is multipled by a constanet and then added (or subtracted) from another row.

Question

Consider the matrix $A= \left[\begin{matrix} 1 & 3 \\ 0 & 2 \end{matrix}\right]$. What operators can you use to put the above equation into it’s reduced row echelon form?