40.1: Linear Systems

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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.

Multiply any row of a matrix by a constant
Add the contents of one row by another row.
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?