Originally Posted by

**Jskid** Find a basis for and the dimensions of the solution space of the given homogeneous system.

$\displaystyle x_1-x_2+2x_3+3x_4+4x_5=0$

$\displaystyle -x_1+2x_2+3x_3+4x_4+5x_5=0$

$\displaystyle x_1-x_2+3x_3+5x_4+6x_5=0$

$\displaystyle 3x_1-4x_2+1x_3+2x_4+3x_5=0$

So I make a matrix with these coefficients and rref to give

$\displaystyle

\[

\left[ {\begin{array}{ccccc}

1 & 0 & 0 & 0 & \frac{1}{3} \\

0 & 1 & 0 & 0 & 0 \\

0 & 0 & 1 & 0 & \frac{4}{3} \\

0 & 0 & 0 & 1 & \frac{1}{3} \\

\end{array} } \right]

\]$

(Here's the part I'm not sure if I did right)

Every solution is of the form $\displaystyle

\[

\left[ {\begin{array}{c}

\frac{-1}{3}r \\

0 \\

\frac{-4}{3}r \\

\frac{-1}{3}r \\

\end{array} } \right]

\]$ where r is any real number. The dimension is 1.