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.