This abbreviated pset is due at 11am on Friday Sep 30.
(a) Give the rank of $D$ and a basis for the nullspace $N(D)$ for the $D$ matrix from pset 1: $$ D=\begin{pmatrix} -\frac{1}{\Delta x} & \frac{1}{\Delta x} & 0 & 0 & 0\\ 0 & -\frac{1}{\Delta x} & \frac{1}{\Delta x} & 0 & 0 \\ 0 & 0 & -\frac{1}{\Delta x} & \frac{1}{\Delta x} & 0 \\ 0 & 0 & 0 & -\frac{1}{\Delta x} & \frac{1}{\Delta x} \end{pmatrix} $$ (Does your answer depend on the value of $\Delta x$?)
(b) Consider the vector space $V$ consisting of all differentiable functions $f(x)$, and let $A = \frac{d}{dx}$ be the linear operator that takes the derivative, i.e. $Af = f'$ for any $f \in V$. Give a basis for $N(A)$, the nullspace of the derivative. Is there a resemblance to your answer in part (a)?
In class, we considered elimination on the matrix $A = \begin{pmatrix} 1 & 2 & 3 & 1 \\ 1 & 2 & 5 & -3 \\ 1 & 2 & 7 & -7 \end{pmatrix}$, and found that it was rank 2 with the pivot and free columns "interleaved" (the first and third columns were pivot columns).
(a) Give a permutation matrix $P$ such that doing Gaussian elimination on $AP$ (which re-orders the _______ of $A$) leads to the first two columns being the pivot columns (i.e. "non-interleaved" pivot and free columns).
(b) How do the nullspace $N(AP)$ and column space $C(AP)$ relate to the null and column spaces of $A$, respectively? In class, a basis for $N(A)$ was the two special solutions $[-2,1,0,0]$ and $[-7,0,2,1]$ and from these how can you get a basis for $N(AP)$? (Note: the problem gave the nullspace vector as $[-7,0,3,1]$, following a typo from the notes. You won't be penalized if you copy this and use 3 instead of 2 here.)
In class, we saw that invertible row operations (such as those occuring in Gaussian elimination) do not change the null space of a matrix.
(a) What happens if you do non-invertible row operations $B = EA$? Does that generally lead to $N(B) = N(A)$, or $N(B) \subseteq N(A)$ (a smaller nullspace), or $N(B) \supseteq N(A)$ (a larger nullspace), or none of these (neither nullspace contains the other)? Justify your answer.
(b) Illustrate your answer in (a) with example $A$, $E$, and $B$ matrices, and give bases for the nullspaces of your $A$ and $B$.
(from Strang, section 3.3)
Give examples of matrices $A$ for which the number of solutions to $Ax=b$ is
(a) 0 or 1, depending on $b$.
(b) $\infty$, regardless of $b$.
(c) 0 or $\infty$, depending on $b$
(d) 1, regardless of $b$.