\magnification=1250

\input amstex

\documentstyle{amsppt}

\vsize=7.2in

\hsize=5.2in

\nopagenumbers


\topmatter

\title
Math 261 -- Linear Algebra\vskip 10pt
Fall 1999 (Exam 3)
\endtitle


\endtopmatter


\vskip 20pt
\noindent (1) Give an example of an invertible linear transformation 
from $\Bbb R^3$ into
itself, other than the identity, which has a two-dimensional 
invariant subspace.

\vskip20pt

\noindent (2) a) Let $T:\Bbb R^m\to \Bbb R^n$ be a linear map, where 
$m<n$. Show that $T$ is not
surjective.
b) Restate a) as a theorem about non-homogeneous linear systems of
equations.

\vskip20pt
\noindent (3) Let $T:\Bbb R^2\to \Bbb R^2$ be the linear map given by 
$T(x,y)=(2x+y, x-3y)$.
Let $\beta$ be the canonical basis of $\Bbb R^2$ and let 
$\beta'=\{(1,1), (1,2)\}$. Use theorem
2.23 to find $[T]_{\beta'}$.
\vskip20pt

\noindent (4) a) Consider the linear map $T:\Bbb R^2\to \Bbb R^2$ 
which consists in rotating
counterclockwise around the origin by $\pi
\over 2$. Write down the matrix of $T$ relative to the canonical basis.
\newline
  b) What are the invariant subspaces of $T$?

\vskip20pt
\noindent (5) Let $V$ be a finite dimensional vector space and $W$ a 
proper subspace of $V$.
Let $a$ be in the complement of $W$ in $V$. Show that there is a 
linear functional $v^{\ast}
\in V^{\ast}$ such that $v^{\ast} (a)=1$ and $v^{\ast}(w)=0$ for all 
$w\in W$. Hint:
complete a basis for $\Cal L\{W,a\}$ to a basis of $V$ and then 
consider its dual.


\end
