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I am looking at the solutions to homework 4, problem 4.8a in Stephen Boyds online course here: https://see.stanford.edu/Course/EE364A/78

The problem is about minimizing a linear objective:

$min \,\, c^Tx, \\subject \,\, to \,\,Ax=b$

And, in the solutions ( which can be found here https://see.stanford.edu/materials/lsocoee364a/hw3sol.pdf) for problem 4.8a, it says "and $c$ is orthogonal to the nullspace of $A$. We can decompose c as $c=A^T\lambda+\hat c, \,\, A\hat c=0.$"

So, from what I am seeing in that solution, I can deduce the following information:

  • $\hat c$ is in the nullspace of A
  • $A^T\lambda$ is in the rowspace of A
  • we are adding $\hat c$ and $A^T\lambda$ to get vector $c$, which is apparently orthogonal to the nullspace of $A$.

If $A$ is square, then this can work. But, if $A$ is not square, then I do not understand how adding a vector in the nullspace and a vector in the rowspace is possible. They are different dimensions, right?

Furthermore, if someone can help me understand how it is possible for $c$ to be orthogonal to the nullspace, and also be in the rowspace of A, that would be much appreciated.

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    $\begingroup$ For $A\in\mathbb{R}^{m\times n}$, both the nullspace and row space of $A$ are subspaces of $\mathbb{R}^{n}$, so they both contain vectors of length $n$ $\endgroup$ – nemo Sep 16 '17 at 18:18
  • $\begingroup$ I guess that makes sense. Can you tell me how $c$ is orthogonal to the nullspace here? $\endgroup$ – nundo Sep 16 '17 at 18:23
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Let $A\in\mathbb{R}^{m\times n}$. For any $v\in\mathbb{R}^{n}$ and a subspace $S$ of $\mathbb{R}^{n}$, $v$ can be written as the sum of its projection onto $S$ and the complement $S^{\perp}$, i.e., \begin{equation} v=P_{S}v+P_{S^{\perp}}v. \end{equation}

In the reference you mention, $S$ is the nullspace of $A$ and $v=c$. So, the first component is something in the nullspace of $A$, let's call it $\hat{c}$ (so $A\hat{c}=0$). The second component is something perpendicular to the nullspace of $A$, or equivalently, something in the rowspace of $A$, so it can be represented as $A^{\top}\lambda$ for some $\lambda\in\mathbb{R}^{m}$. So we have \begin{equation} c=A^{\top}\lambda+\hat{c}. \end{equation}

When $c$ is orthogonal to the nullspace of $A$, $\hat{c}=0$, and \begin{equation} c=A^{\top}\lambda\implies c^{\top}x=\lambda^{\top}Ax=\lambda^{\top}b. \end{equation}

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  • $\begingroup$ This is awesome!! Thank you $\endgroup$ – nundo Sep 16 '17 at 20:24
  • $\begingroup$ I understand now, the trick is that when the vector $c$ is orthogonal to the sum of two orthogonal components, then the component in the nullspace MUST be $0$ by the definition of something being orthogonal. $\endgroup$ – nundo Sep 16 '17 at 21:08
  • $\begingroup$ now, @nemo, the theorem you stated at the beginning - is that only true for $v \in \mathbb{R}^n$? Or is it true for a member of any vectorspace? i.e. is it true that, for any vector space, a member of it can be decomposed into a projection onto a subspace of that vector space, plus a projection onto the orthogonal completment of that subspace? $\endgroup$ – nundo Sep 17 '17 at 17:16
  • $\begingroup$ @nundo This kind of decomposition can be done for any vector space that is equipped with an inner product structure; for more details you can read this for a start $\endgroup$ – nemo Sep 18 '17 at 4:51

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