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\centerline{Mathematics 244, section 1 -- Linear Algebra}
\centerline{Exam 3 -- April 30, 2004}
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{\it Directions}
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Do all work in the blue exam booklet.  There are 100 regular points
and 10 Extra Credit points.    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.
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\noindent
I. (25) Let $T : {\bf R}^3 \to {\bf R}^3$ be the linear mapping whose
matrix with respect to the standard basis is 
$$A = \pmatrix{1&3&-1\cr
               2&0&4\cr
               1&-2&1\cr}$$
Find the matrix $[T]_\beta^\beta$, where
$$\beta = \{(1,1,1),(1,1,0),(1,0,0)\}$$
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\noindent
II. (20)  Let $V$ be a finite-dimenional vector 
space, let $T : V \to V$ be a linear mapping.
Show by mathematical 
induction that if ${\bf x}_i\in V$ are vectors 
satisfying $T({\bf x}_i) = \lambda_i {\bf x}_i$ for 
$i = 1, \ldots, n$, where $\lambda_i$ are distinct real numbers, then
$\{{\bf x}_1,\ldots, {\bf x}_n\}$ is linearly independent.
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III.  Let $T: P_2({\bf R}) \to P_2({\bf R})$ be the linear
mapping defined by 
$$T(p(x)) = 2xp'(x) + (x+1)p''(x).$$
\item{A)}  (20) Find the matrix of $T$ with respect
to the basis $\{1,x,x^2\}$ of $P_2({\bf R})$ and
the eigenvalues of $T$.  For each eigenvalue, find a basis for the 
corresponding eigenspace (express as polynomials).
\item{B)} (5) Is $T$ diagonalizable?  Why or why not?
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\noindent
IV.  (True-False)  For each true statement, give a short proof.
For each false statement, give a reason or a counterexample.  
\item{A)} (10)  If $A$ is an $n\times n$ matrix whose
rows are linearly dependent, then $\det(A) = 0$.
\item{B)} (10) The vector space $M_{2\times 3}({\bf R})$
is isomorphic to $P_6({\bf R})$.
\item{C)} (10)  If the columns of an $n\times n$ matrix $Q$ form an 
orthonormal basis for ${\bf R}^n$, then $Q^tQ = I_n$.
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{\it Extra Credit} (10)  If $a,b,c$ are distinct real numbers, show
that the matrix 
$$A = \pmatrix{1&a&a^2\cr
               1&b&b^2\cr
               1&c&c^2\cr}$$
is invertible.   
\bye