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1.\ \ Let $T$ be a linear operator on $\Bbb R^3$ defined by

$T(x_1, x_2, x_3) = (x_1 + 2x_2 + 7x_3, 2x_1 - 2x_2 + 2x_3, 6x_2 + 12x_3)$


\itemitem{a)} What is the matrix of $T$ in the standard ordered basis for $\Bbb
R^3$? 
\vfill
\itemitem{b)} Find a basis for the null space of $T$.
\vfill
\itemitem{c)} Find a basis for the range of $T$.

\newpage

2.\ \ Let $\Bbb V$ be the vector space of all $(3\times 3)$ matrices.  Let $W_0$ and
$W_1$ be subspaces of $V$ defined by 
$$
\aligned
&W_0 = \{A \in \Bbb V|A_{11} + A_{22} + A_{33}= 0\}\\
&W_1 = \{A \in \Bbb V|A_{ij} = A_{ji} \text{ for all } i,j\}
\endaligned
$$


\itemitem{a)}Find the dimension and a basis for $W_0$.
\vfill
\itemitem{b)}Find the dimension and a basis for $W_1$.
\vfill
\itemitem{c)} Find the dimension and a basis for $W_0 \cap W_1$.
\vfill
\itemitem{d)} Find the dimension and a basis for $W_0 + W_1$.

\newpage

3.\ a)\ Let $A$ and $B$ be $n\times n$ matrices.  Show that $Tr\ AB = Tr\ BA$
\vfill
\itemitem{b)} Show that similar matrices have the same trace.
\vfill
\itemitem{c)}Prove that the matrices $A = \left[\matrix 1&0\\2&3 \endmatrix
\right]\ B = \left[\matrix 1&0\\3&2\endmatrix \right]$ are not similar. \vfill
\itemitem{d)} Let $A$ be the same as in c).  What is $A^t$?  Are $A$ and $A^t$
similar?

\newpage 

4.\ \ Determine which of the following subsets of $\Bbb F[x]$ are ideals.  If it is,
find its monic generator. 
\itemitem{a)}$\{f|f(0) = f'(0) = 0\}$
\vfill
\itemitem{b)}$\{f = (x^2 + 3x + 2) g + (x^3 + 2x^2 + x + 2)h|\ g, h \in \Bbb F[x]\}$

\newpage

\itemitem{c)}$\{f| deg f \ge 1\}$
\vfill
\itemitem{d)} $\{f| f(A) = 0$, where $A = \left[\matrix2&0\\-2&1\endmatrix \right]\}$

\newpage

5.\ \ Let $\Cal P_3$ be the space of all polynomials of degree less or equal to $3$. 
Let $D$ be the differential operator on $\Cal P_3$ defined by
$$D(\sum^3_{k=0} C_k x^k) = \sum^3_{k=1} k C_k x^{k-1}$$

\itemitem{a)}Is $D$ invertible?  What are the null space $N_D$ and the range $R_D$
of $D$?  Find a basis for $N_D$ and $R_D$ respectively. \vfill
\itemitem{b)} What is the respresentation of $D$ with respect to the standard basis
\newline
$\Cal B = \{1, x, x^2, x^3\}$?
\vfill
\itemitem{c)} Let $\Cal B^\ast$ be the dual basis for $\Cal P^\ast_3$.  What is the
representation for $D^t$, the transpose of $D$ with respect to $\Cal B^\ast$?

\newpage

\itemitem{d)} Let $f \in \Cal P_3^\ast$ be the linear functional defined by
$$f(p) = \int^1_0 p(x) dx\ \ \ \ p(x) \in \Cal P_3$$

What is $D^tf(p)$ where $p(x) = x^2 + 1$?
\vfill
\itemitem{e)}  Let $\Cal B' = \{1, (x + 1), (x + 1)^2, (x + 1)^3\}$.  Show that
$\Cal B'$ is a basis for $\Cal P_3$.  What is the representation for $D$ with
respect to $\Cal B'$? 
                                                                                                                                               
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