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I'm reading a textbook, and author says
"Any vector $\hat{x}$ can be split into a row space component $x_r$ and a nullspace component $x_n$: $\hat{x}=x_r+x_p$

I guess it represents the "Complete solution": $x=x_p+x_n$
Is that right? If so, the particular solution $x_p$ is in row space?

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    Sorry, I omitted too many things. Complete solution is: $Ax_p=b$ and $Ax_n=0$ produce $A(x_p+x_n)=0$.2019-02-18

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So, I suppose we have a matrix $A\in\Bbb R^{N\times M}$, and we want to solve $Ax=b$ Then, indeed, if $x_p$ is any solution (named 'particular'), then $A(x-x_p)=0$ is remained to solve, hence in this case, all the solutions are of the form $x_p+x_n$ as you write, where $x_n$ is in the nullspace of $A$.

On the other hand, according to the textbook, every vector $x$ can be written as $x_r+x_n$ where $x_r$ is in the rowspace of $A$ and $x_n$ is in the nullspace. (For this, consider the rowspace and project $x$ orthogonally to it.)

The common point in the above ones is the 'nullspace', and thus we can state not more than

An $x$ is a solution iff its row space component, $x_r$ is a solution.