This abbreviated pset is due at 11am on Wednesday Feb 23. The material on this pset may be on exam 1 (Feb. 25).
(a) Does $C(A)$ necessarily contain the $C(AB)$ or vice versa? (That is, is $C(A) \subseteq C(AB)$ or $C(AB) \subseteq C(A)$?)
(b) Give an example of a square matrix $A$ where $C(A^2)$ is lower-dimensional than $C(A)$.
(c) If $V$ is the set of polynomials $f(x)$ of degree $< 4$, consider the derivative linear operator $d/dx$ acting on $V$. Give a basis for $N(d/dx)$ and $C(d/dx)$, the null and column spaces of this operator for inputs in $V$.
Come up with a matrix $A$ and a vector $b\ne 0$ such that the solutions $x$ of $Ax=b$ form a line in $\mathbb{R}^3$, where all of the entries of $A$ are nonzero. Find the complete solution (i.e., all solutions).
(i.e. come up with your own homework problem — how do you think Gil Strang does it?)
The following matrix is from problem 7(a) of pset 2. Feel free to re-use the pset–2 solutions as needed. $$ A = \begin{pmatrix} 1 & 2 & 3 & 4 \\ 1 & 2 & 4 & 6 \\ 0 & 0 & 1 & 2 \end{pmatrix} $$
(a) Give the dimensionality and a basis for $C(A)$.
(b) If $b = \begin{pmatrix} \alpha \\ 6 \\ 1 \end{pmatrix}$, for what values of $\alpha$ will $Ax=b$ have a solution?
(c) For the $\alpha$ from (b), give the complete solution to $Ax=b$.