# Eigenspaces homework problem

## Homework Statement

Find the Eigenspace of the following matrix:
$$\begin{bmatrix} 1 & 3 \\ 4 & -3 \end {bmatrix}$$
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:
$$\begin{bmatrix} -6 & -3 \\ -4 & -2 \end {bmatrix}$$
Then you find the null space, which starts with putting it in reduced row echelon form:
$$\begin{bmatrix} -6 & -3 \\ 0 & 0 \end {bmatrix}$$
you can reduce that further to
$$\begin{bmatrix} -2 & -1 \\ 0 & 0 \end {bmatrix}$$
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
$$\begin{bmatrix} -1 \\ 2 \end {bmatrix}$$
My textbook confirms that. How do I get from here
$$\begin{bmatrix} -2 & -1 \\ 0 & 0 \end {bmatrix}$$
to there
$$\begin{bmatrix} -1 \\ 2 \end {bmatrix}$$
I would think you would just eliminate the 2nd row and transpose the first row but that would give.
$$\begin{bmatrix} -2 \\ -1 \end {bmatrix}$$

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## Answers and Replies

LCKurtz
Science Advisor
Homework Helper
Gold Member

you can reduce that further to
$$\begin{bmatrix} -2 & -1 \\ 0 & 0 \end {bmatrix}$$
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
$$\begin{bmatrix} -1 \\ 2 \end {bmatrix}$$
My textbook confirms that. How do I get from here
$$\begin{bmatrix} -2 & -1 \\ 0 & 0 \end {bmatrix}$$
to there
$$\begin{bmatrix} -1 \\ 2 \end {bmatrix}$$
I would think you would just eliminate the 2nd row and transpose the first row but that would give.
$$\begin{bmatrix} -2 \\ -1 \end {bmatrix}$$

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.