Pset A (Required More Substantial Pset, not droppable) Due Wednesday April 22. You may use all available materials. Some collaboration allowed but do your own work.

(1) Compute the projection matrices $\frac{aa^T}{a^Ta}$ onto the lines through $a_1=(-1,2,2)\in R^3$ and $a_2=(2,2,-1)\in R^3$. Multiply those projection matrices and explain why their product $P_1P_2$ is what it is.

(2) A is an $m$ by $n$ matrix of rank $r$. Suppose there are right hand sides $b$ for which $Ax=b$ has no solution.

(a) What are all inequalities ($<$ or $\le$) that must be true between $m,n$ and $r$?
(b) How do you know that $A^T y=0$ has solutions other than $y=0$?

(3) Show that if $Q$ is nxn orthogonal, $\det(Q)=\pm 1$

(4) We have

using LinearAlgebra
U, = qr(rand(3,3))
V, = qr(rand(3,3))
D  = Diagonal([3,2,1])
A = U*D*V'

3×3 Array{Float64,2}:
 2.19565   -0.00886362  -0.386611
 1.03165    1.18653     -0.776745
 0.889864   1.95349      1.16022

We have $\det(A)=\pm$(what number?) Why?

(5) $Q$ is a tall skinny 3x2 matrix. ($Q^TQ=I$) A picture of a corgi sits on the unit square in $R^2$. Describe carefully but briefly the image of this picture under $Q$. (What is the shape? What is the area? Where are the vertices?)

(6) A solid box has volume 10 cubic meters. What is the shape of the image of the box under the transformation $A$ in question (4) above. What is the volume of this image?

(7) Suppose $A=U\Sigma V^T$ is the rank-r svd of an mxn matrix $A$. Write $x_r=$ the projection of x onto the rowspace of $A$ in terms of possibly $x$, $U$, or $V$.

(8) As in (7), write $x_n$=projection of $x$ on the nullspace of $A$ in terms of possibly $x$,$U$,$V$.

(9) As above, what is $x_r+x_n$ in simple terms?

(10)As above, suppose $Ax_r=b$, what is $Ax$ in simplest form?

(11) As above, if $Ax_r=b$, is $b$ in the column space of $U$? Why or why not?

(12) Using the rank-r SVD, if $b$ is in col($A$), what is the unique solution in row($A$) to $Ax=b$?

(13) Use the fact that swapping two rows of a matrix $A$ flips the sign of a determinant to show that det($A$)=0 if $A$ has two equal rows.

(14) Am I a linear transformation? (A linear transformation is a function $f$ from one vector space to another that satisfies $f(c_1x_1+c_2x_2)=c_1f(x_1)+c_2 f(x_2)$.

(14a) $f(A)=trace(A)$ from $n\times n$ A to $R$
(14b) $f(A)=\det(A)$
(14c) $f(x)=c^Tx$ for $x \in R^n$ (c constant in $R^n$)
(14d) $f(A)=M^TA$, for $m \times n$ $A$ and constant $M$ with $m$ rows.
(14e) $f(A)=X^T A Y$ for $m \times n$ $A$ and compatible constant matrices $X$ and $Y$
(14f) $f(A) = A^TA$ for $m \times n$ $A$
(14g) $f(A)= A+A^T$ for $n \times n$ $A$

(15) Suppose $m \times n$ $B$ has rank $m$. Let $B = U\Sigma V^T$ be the rank-r svd. What is $(BB^T)^{-1}?$