What is the Eigenspace for a 2x2 matrix with eigenvalues -5 and 3?

[g.lemaitre]

Homework Statement

Find the Eigenspace of the following matrix:
\begin{bmatrix}<br /> 1 &amp; 3 \\<br /> 4 &amp; -3<br /> \end {bmatrix}
I'm skipping a few steps but the Eigenvalues are -5 and 3. Let's starts with -5. Skip a few more steps, I know I'm right, just trust me.
We now have the following matrix:
\begin{bmatrix}<br /> -6 &amp; -3 \\<br /> -4 &amp; -2<br /> \end {bmatrix}
Then you find the null space, which starts with putting it in reduced row echelon form:
\begin{bmatrix}<br /> -6 &amp; -3 \\<br /> 0 &amp; 0<br /> \end {bmatrix}
you can reduce that further to
\begin{bmatrix}<br /> -2 &amp; -1 \\<br /> 0 &amp; 0<br /> \end {bmatrix}
This is where I'm confused. This nullspace calculator http://www.math.odu.edu/~bogacki/cgi-bin/lat.cgi
says that the basis of the null space is
\begin{bmatrix}<br /> -1 \\<br /> 2<br /> \end {bmatrix}
My textbook confirms that. How do I get from here
\begin{bmatrix}<br /> -2 &amp; -1 \\<br /> 0 &amp; 0<br /> \end {bmatrix}
to there
\begin{bmatrix}<br /> -1 \\<br /> 2<br /> \end {bmatrix}
I would think you would just eliminate the 2nd row and transpose the first row but that would give.
\begin{bmatrix}<br /> -2 \\<br /> -1<br /> \end {bmatrix}

 

[LCKurtz]

you can reduce that further to
\begin{bmatrix}<br /> -2 &amp; -1 \\<br /> 0 &amp; 0<br /> \end {bmatrix}
This is where I'm confused. This nullspace calculator http://www.math.odu.edu/~bogacki/cgi-bin/lat.cgi
says that the basis of the null space is
\begin{bmatrix}<br /> -1 \\<br /> 2<br /> \end {bmatrix}
My textbook confirms that. How do I get from here
\begin{bmatrix}<br /> -2 &amp; -1 \\<br /> 0 &amp; 0<br /> \end {bmatrix}
to there
\begin{bmatrix}<br /> -1 \\<br /> 2<br /> \end {bmatrix}
I would think you would just eliminate the 2nd row and transpose the first row but that would give.
\begin{bmatrix}<br /> -2 \\<br /> -1<br /> \end {bmatrix}

The matrix corresponds to -2x - y = 0 or y = -2x So$$
\begin{bmatrix}x\\ y \end{bmatrix}=\begin{bmatrix}x\\ -2x \end{bmatrix}
=x\begin{bmatrix}1\\ -2 \end{bmatrix} $$
Any nonzero constant works, so take $x=-1$ to get their answer.