"Let $A$ be a $4 \times 4$ random matrix with rank $2$ (check that its rank is $2$). Let $b$ be a random vector in $\mathbb R^4$.
Check that the system $Ax = b$ is inconsistent and then find a least squares solution $x_0$ of $Ax = b$ (is this solution unique?).
Compute the error vector $b-Ax_0$. Check that this error vector is perpendicular to the
column space of $A$.
Compute the error is the length $\|b – Ax_0\|$. Check that this error is minimized for the
least squares solution by computing $\|b – Ax\|$ for several random vectors $x$ and seeing that it is larger than the error."
So far I have:
A = rand(4,2)*rand(2,4); rank(A); b=rand(4,1); C=inv(A)*b; x = (inv(A'*A))*A'*b; E = b-A*x; D = norm(E);
The first three lines define $A$ and $b$ and prove the rank is 2.
C is intended to prove $Ax=b$ is inconsistent (it returns a strange answer).
x should be the least squares solution, this apparently is not unique because of the last question, but I don't know how else to find an
x. E is the error vector; I do not know how to check it's perpendicular to the column space of $A$.
D is the length of the error vector; I do not know how to find other
x's to test.