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{\bf \centerline
{Math 262 -- Linear Algebra}} 
{\bf \centerline{Spring 2001 (Exam 1)}}
\bigskip
\noindent{\bf 1.} {\sl (25 points)} Find a $3 \times 3$ matrix $A$ over the reals so that
(i)--(iii) below all hold:

{(i)} $1$ is an eigenvalue of $A$, with eigenvector $(0,0,1)$,

{(ii)} $5$ is an eigenvalue of $A$, with eigenvector $(\frac{3}{5},
\frac{4}{5}, 0)$, and
 
{(iii)} $-10$ is an eigenvalue of $A$, with eigenvector $(\frac{4}{5},
-\frac{3}{5}, 0)$.

\bigskip
{\noindent \bf 2.} {\sl (25 points)}   Consider the linear operator $T:\Bbb R^3 \to \Bbb
R^3$ given by  
$$T(x,y,z)=(x, x+2y+3z, x+2z).$$


\noindent a) Write down the matrix (with respect to the standard ordered basis) of $T$
and determine the
characteristic polynomial of
$T$.
\smallskip
\noindent b) Find the eigenvalues of $T$.
\smallskip
\noindent c) Find a basis for  the eigenspace of $T$ corresponding to the \it 
double \rm eigenvalue.
\smallskip
\noindent d) Is $T$ diagonalizable? Explain.




\bigskip


\bigskip 

\noindent{\bf 3.} {\sl (25 points)} Prove the \it {parallelogram law} \rm on an inner product
space 
$V$; that is, show that for all vectors $x$ and $y$ in $V$
$$||x+y||^2+||x-y||^2=2||x||^2+2||y||^2.$$
What does this equality say about parallelograms in $\Bbb R^2$?

\bigskip
\noindent{\bf 4.} {\sl (25 points)} Consider the real inner product space $V$ consisting of
polynomials of degree at most two over the real numbers with inner product 
$\langle f,g\rangle =\int_0^1 f(x)g(x)dx$.
\smallskip
\noindent a) Apply the Gram-Schmidt process to the ordered set of vectors $X=\{x,x^2\}$
in $V$. Write down an orthonormal basis for the span $W$  of $X$.
\smallskip
\noindent b) Determine the orthogonal projection of the constant function $1$ on $W$.    
\smallskip
\noindent c) Find a basis for the orthogonal complement $W^\perp$ of $W$.
%\noindent{\bf 5.} {\sl (20 points)} Consider the space $M_{n\times n} (\Bbb R)$ of all
%$n\times n$  real matrices endowed with the inner product $\langle
%A,B\rangle=\text{tr}(B^{\ast}A)$ (recall the trace $\text{tr}(X)$ of a sqaure $n\times
%n$ matrix
%$X$
% is the sum $\sum_{i=1}^n X_{ii}$ of its diagonal entries). Consider the subset $W$ of
%$M_{n\times n} (\Bbb R)$ consisting of those  matrices of zero trace. You may assume that $W$
%is a subspace of dimension 
%$n^2-1$. Find the orthogonal
%complement $W^\perp$.




\vskip30pt

%2) Show by means of an example that the conclusion in problem 1 may 
%fail if $V$ is a real vector space.
%
%\vskip30pt
%
%3)  Consider the operator $T:\Bbb C^4\to \Bbb C^4$ given by 
%$$T(z_1,z_2,z_3,z_4)=(z_1+iz_2+z_3+z_4,z_2,z_2+3z_4, z_2+4z_4).$$
%\noindent
%Is $T$ diagonalizable? What is the minimal polynomial of $T$?
%
%
%

\vfill\eject\bye
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