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- #1

Let's say, the following matrix is reduced.

From:

$\begin{bmatrix}0 & 2 & -8\\0 & 2 & 0\\0 & 0 & 2\end{bmatrix}$

To:

$\begin{bmatrix}0 & 1 & 0\\0 & 0 & 1\\0 & 0 & 0\end{bmatrix}$

To Parametrize that I would do the following:

x2=0, x3=0

$\begin{bmatrix}x1\\x2\\x3\end{bmatrix}$= $\begin{bmatrix}0\\1\\1\end{bmatrix}$

But that doesn't seem right. For some reason when the matrix is bigger with more integers I can do it simply with chosen variables r, s, t... But with only zeroes like this I get super confused.

I would usually proceed with assigned variables, which may yield:

$\begin{bmatrix}x1\\x2\\x3\end{bmatrix}$= r $\begin{bmatrix}0\\1\\0\end{bmatrix}$ + s $\begin{bmatrix}0\\0\\1\end{bmatrix}$

But even this feels odd, seeing as I'm assigning a variable to a pivot.

Can someone please clarify this for me.

Thank You.