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The question is :

Find a 2 by 3 system $Ax=b$ whose complete solution is :

$$ x=\begin{bmatrix} 1 \\ 2 \\ 0 \\ \end{bmatrix}+w \begin{bmatrix} 1 \\ 3\\ 0\\ \end{bmatrix} $$

So I treated this as $x=x_p+x_n$ where the 2nd matrix is $x_n$

So I said to myself find a matrix such that it's nullspace is $\begin{bmatrix} 1 \\ 3\\ 0\\ \end{bmatrix} $ I found one but how can it relate it to the particular solution. I know that particular solution would be in the row space of A. But I couldn't manage to find a matrix which contains $ \begin{bmatrix} 1 \\ 2 \\ 0 \\ \end{bmatrix}$ in it's rowspace and $\begin{bmatrix} 1 \\ 3\\ 0\\ \end{bmatrix} $ in it's nullspace at the same time. What to do here, Help me out please.

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  • $\begingroup$ I dont think that the solution is unique. Are you required to find one possible solution? $\endgroup$ Commented Jul 19, 2016 at 17:58

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You don't have to pick just the matrix, but also the right hand side! So, if you've found a matrix $A$ with the given null space, just let $b=Ax_p$. With this choice

$$ b=Ax_p = Ax_p + Ax_n=Ax. $$

Edit: by the way, you cannot find a matrix whose null space contains $[1\ 3\ 0]$ and its row space contains $[1\ 2\ 0]$, since row space and null psace are orthogonal, and those two vectors are not orthogonal.

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  • $\begingroup$ This is more of a comment. $\endgroup$ Commented Jul 19, 2016 at 15:13
  • $\begingroup$ Why? Doesn't it answer the question of finding a system with the given solution expression? $\endgroup$ Commented Jul 19, 2016 at 15:14
  • $\begingroup$ doesn't answer my question, would have been better with an example (example could be taken as my question) so. Would appreciate if you edit your answer $\endgroup$ Commented Jul 19, 2016 at 15:21
  • $\begingroup$ Hum, yes, it does. The question asks to find a 2x3 system with the given general solution. You said you found a matrix with the right null space (great!), and I explained how to find the missing part (the right hand side). Then, in the edit, I also explained why $[1\ 2\ 0]$ cannot be in the row space. How is this not an answer to your question? $\endgroup$ Commented Jul 19, 2016 at 15:35
  • $\begingroup$ Doesn't A transform particular solutions(from the row space) to the column space? So if 1 2 0 isn't in the rowspace since collapse has 1 3 0 how will I find the matrix A? $\endgroup$ Commented Jul 19, 2016 at 15:40

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