LINEAR ALGEBRA (Math 211)
HW #16 (Due Mon. Dec. 4 )


1. Let T1 and T2 be the reflections of ${\mathbb R}^2$ about the lines y=0 and $y=-\sqrt{3}x$. How many linear operators can be obtained from T1 and T2 using all possible compositions? Find the matrices of all of them.


2. Find the dimensions and bases for the kernel and for the image of the following transformations
$\textstyle \parbox{5cm}{ \begin{tabular}{cl} (a)&$T:{\mathbb R}^3\to{\mathbb R}^2$\\ &$T(x,y,z)=(z,y)$\end{tabular}}$                 $\textstyle \parbox{4.3cm}{ \begin{tabular}{cl} (b)&$T:{\mathbb R}^2\to{\mathbb R}$\\ &$T(x,y)=x+y$\ \end{tabular}}$                 $\textstyle \parbox{4.4cm}{ \begin{tabular}{cl} (c)&$T:P_2\to{\mathbb R}$\\ &$T(P(x))=P(1)$, \end{tabular}}$


where P2={P(x)= ax2+bx+c} is the space of quadratic polynomials.


3. Find a linear transformation T from ${\mathbb R}^3\to{\mathbb R}^3$ such that

\begin{displaymath}\mbox{(a)}\quad \ker T=\{(x,y,z):2x-y+z=0\}\qquad\qquad\qquad \mbox{(b)}\quad \mbox{im\ } T=\{(x,y,z):2x-y+z=0\}\end{displaymath}


4. A linear operator $T:{\mathbb R}^2\to{\mathbb R}^2$ has the matrix $\left(\begin{array}{cc} 2&1\\ 0&-1 \end{array}\right)$ with respect to the standard basis for ${\mathbb R}^2$. Find the matrix of this operator with respect to the basis $\mbox{\bf e}'_1=(2,-1)$, $\mbox{\bf e}'_2=(-1,1)$


5. A linear operator $T:{\mathbb R}^3\to{\mathbb R}^3$ has the matrix $\left(\begin{array}{ccc} 1&0&0\\ 0&-1&0\\ 0&0&2 \end{array}\right)$ with respect to the standard basis for ${\mathbb R}^3$. Find the matrix of this operator with respect to the basis $\mbox{\bf e}'_1=(2,0,3)$, $\mbox{\bf e}'_2=(4,1,5)$, $\mbox{\bf e}'_2=(3,-1,7)$.