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{\bf \centerline{Math 261 -- Linear Algebra}}\vskip 10pt
{\bf \centerline{ Fall 2000 (Final Exam)}}
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\noindent {\bf 1.} {\sl(30 points)} Consider the linear transformation $T\colon \bf R^4\rightarrow \bf R^3$
given by
$$T(x_1,x_2,x_3,x_4)=(3x_1-2x_2+4x_3-x_4,2x_1+3x_2-5x_3+4x_4,4x_1-7x_2+13x_3-6x_4).$$
\smallskip  
 (a) Determine a basis for the null space of $T$.\smallskip
(b) Determine a basis for the range of $T$.

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\noindent {\bf 2.} {\sl (30 points)} Explicitly find an ordered basis $\beta$ for $\bf R^2$  such
that the matrix
$[T]_{\beta}^\beta$ of the linear transformation $T:\bf R^2\to \bf R^2$ given by $T(x,y)=(x+2y, 
2x+y)$ is a diagonal matrix.

\vskip20pt

\noindent {\bf 3.} {\sl (30 points)} Let $V$ be a finite-dimensional vector space.
\smallskip (a) Explain what is meant by
saying that  linear maps
$S,T:V\to  V$ are similar.
\smallskip (b) Prove that similar linear maps $S,T:V\to  V$  have the same rank.

\vskip20pt

\noindent {\bf 4.} {\sl(30 points)}\smallskip(a) Let $V$, $W$ be finite-dimensional vector spaces over a
field
$F$. Prove that there is a {\sl surjective} linear map $T\colon V\rightarrow W$ if and only
if
$\hbox{\rm dim}(V)\geq
\hbox{\rm dim}(W)$.
\smallskip (b) Is there an
{\sl injective } linear  transformation from the space of all
$n\times n$ symmetric real matrices into $\bf R^{2n}$ ? 

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\noindent{\bf 5.} {\sl (30 points)} \smallskip
(a) Define the determinant of an $n\times n$ matrix over a field $F$. \smallskip
(b) Let $A^{(n)}$ be the $n\times n$ real matrix
with $2$'s on the main diagonal, $1$'s on the diagonals above and below the main diagonal and zero's
everywhere else i.e with
 entries
$(A^{(n)})_{i,i}=2$,
$(A^{(n)})_{i,j}=1$ if $j-i=\pm 1$ and $(A^{(n)})_{i,j}=0$ otherwise.
Prove that $\det(A^{(n)})=n+1$. (Hint:  Let $a_n:=\hbox{\rm det} (A^{(n)})$.
 Show by expansion along the
first row that 
$a_n=2a_{n-1}-a_{n-2}$ for
$n\geq 3$).
 


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