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\centerline{Mathematics 244, section 1 -- Linear Algebra}
\centerline{Exam 3 Sample Questions -- April 20, 2007}
\bigskip
\noindent
{\it Notation}
\item{$\bullet$}  $P_n({\bf R})$ is the vector space of 
polynomials of degree $\le n$.
\item{$\bullet$} $P({\bf R})$ is the vector space
of all polynomials (no restriction on degree).
\item{$\bullet$} $M_{m\times n}({\bf R})$ is the vector
space of all $m\times n$ matrices.
\bigskip
\noindent
I. Let $T : {\bf R}^3 \to {\bf R}^3$ be the linear mapping whose
matrix with respect to the standard basis is 
$$[T]_{std \leftarrow std} = \pmatrix{1&3&-1\cr
               2&0&4\cr
               1&-2&1\cr}$$
Let 
$${\cal B} = \left\{\pmatrix{1\cr 1\cr 1\cr}, \pmatrix{1\cr 1\cr 0\cr},
\pmatrix{1\cr 0\cr 0\cr} \right\} \quad {\rm and} \quad
{\cal C} = \left\{\pmatrix{2\cr -1\cr 3\cr}, \pmatrix{0\cr -1\cr 3\cr},
\pmatrix{2\cr 1\cr 0\cr} \right\}$$ 
be bases for ${\bf R}^3$.
\item{A)}  What is the change of coordinates matrix ${\cal P}_{C \leftarrow std}$?
\item{B)}  What is the matrix $[T]_{std \leftarrow {\cal B}}$?
\item{C)}  What is the matrix $[T]_{{\cal C} \leftarrow std}$?
\item{D)}  What is the matrix $[T]_{{\cal B} \leftarrow {\cal C}}$?
\bigskip
\noindent
II. Let $V$ be a finite-dimenional vector 
space, and let $T : V \to V$ be a linear mapping.
Show by mathematical 
induction that if ${\bf v}_i\in V$ are vectors 
satisfying $T({\bf v}_i) = \lambda_i {\bf v}_i$ for 
$i = 1, \ldots, n$, where $\lambda_i$ are distinct real numbers, then
$\{{\bf v}_1,\ldots, {\bf v}_n\}$ is linearly independent.
\bigskip
\noindent
III.  Let 
$$A = \pmatrix{a & -1 & 0\cr
               0 & 3  & 0\cr
               2 & 2  & -4\cr}$$
\item{A)}  Find the characteristic polynomial of $A$.
\item{B)}  Does ${\rm dim Nul}(A - (-4)I)$ depend on the value
of $a \in {\bf R}$?  Explain.  
\item{C)}  For which values of $a \in {\bf R}$ is the matrix
$A$ diagonalizable?
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\noindent
IV.  Let $T: P_2({\bf R}) \to P_2({\bf R})$ be the 
mapping defined by 
$$T(p(x)) = 2xp'(x) + (x+1)p''(x).$$
\item{A)}  Show that $T$ is linear.
\item{B)}  Find the matrix $[T]_{B \leftarrow B}$ with respect
to the basis $B = \{1,x,x^2\}$ of $P_2({\bf R})$ 
\item{C)}  Determine the the eigenvalues of $T$.  
\item{D)}  For each eigenvalue, find a basis for the 
corresponding eigenspace (express the basis elements as polynomials).
Is $T$ diagonalizable?  Why or why not?
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\noindent
V.  (True-False)  For each true statement, give a short proof.
For each false statement, give a reason or a counterexample.  
\item{A)} If $A$ is an $n\times n$ matrix whose
rows are linearly dependent, then $\lambda = 0$ is an eigenvalue of $A$.
\item{B)} If $A$ is an $8 \times 8$ matrix with 
$\det(A - \lambda I) = (2 - \lambda)^3 (3 - \lambda)^5$,
${\rm rank}(A - 2I) = 6$, and ${\rm rank}(A - 3I) = 4$, 
then it is possible for $A$ 
to be diagonalizable.  
\item{C)} If the columns of an $n\times n$ matrix $Q$ form an 
orthogonal set in ${\bf R}^n$, then $Q^tQ = I$, then $n\times n$
identity matrix.
\item{D)}  If $S$ is an orthogonal set of nonzero vectors in 
${\bf R}^n$ then $S$ is linearly independent.   
\item{E)}  If ${\cal B}$ is a linearly independent set in 
the vector space $M_{3\times 4}({\bf R})$, then ${\cal B}$
contains at most $3 + 4 = 7$ matrices.
\bye
