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Give an example of a matrix A such that $\ker(A)$ is the plane $2x − y + 3z = 0$.

I am not sure where to start, as I know that the $\ker(A)$ is the matrix of the plane, but I don't know how to go backwards from it. Would I need to assume that there's a free variable in the rref(A) and assume it was pulled out of the equation of the plane?

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Let $$A=\begin{pmatrix} 2 & -1 & 3\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{pmatrix}$$

Then $$A\begin{pmatrix} x\\ y\\ z\end{pmatrix}=0$$ implies $$2x-y+3z=0$$

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  • $\begingroup$ Ahhhh, I was totally overthinking it. Thanks! $\endgroup$ – 150005388 May 21 at 3:11
  • $\begingroup$ @150005388 You are welcome. $\endgroup$ – Bach May 21 at 3:12
  • $\begingroup$ Why do you need anything beyond the first row? There’s nothing in the question that constrains the size of the matrix. $\endgroup$ – amd May 21 at 3:51
  • $\begingroup$ @amd You are right. I am just used to considering linear transformations in $n$-dimensional vector space $V$: $V\to V$, so my matrix is a square matrix. $\endgroup$ – Bach May 21 at 3:55

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