2.2 Homogeneous Equations

homogeneous

• Linear equations

  • if its in the form $a_1{x}_1+a_2{x}_2+\cdots +a_n{x}_n=0$

• System

  • if all of its equations are homogeneous

• Examples

  • Ex. 1

x_{1}+x_{2}-3x_{4}+2x_{5}=0, \
x_{3}+5x_{4}-x_{5}=0

\end{align}}$-=>$\begin{bmatrix}
1 & 1 & 0 & -3 & 2 & |0 \
0 & 0 & 1 & 5 & -1 & |0
\end{bmatrix} - We can see that this matrix is already in RREF - leading 1s indicate $x_{1},x_{3}$can be written in terms of $x_{2},x_{4},x_{5}$ (free variables)\begin{align}
x_{1}=-x_{2}+3x_{4}-2x_{5} \
x_{3}=-5x_{4}+x_{5}
\end{align}$-thiscanbewrittenasthesolutionsetas$\begin{bmatrix}
x1 \
x2 \
x3 \
x4 \
x5
\end{bmatrix}$$