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\centerline{Mathematics 244, section 1 -- Linear Algebra}
\centerline{Final Examination}
\centerline{May 10, 2007}
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\noindent
{\it Directions}
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Do problems I - VII, and either VIII or IX in the blue exam book.
If you wish, you may submit solutions for {\it both} VIII and IX for
Extra Credit.
Do not place anything you wish to have considered for credit on the 
exam sheet.  There are 200 regular points and up to 20 
Extra Credit points.  
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\noindent
I.  Let 
$$A = \pmatrix{-1 & 0 & 2  & 2 & -1\cr
                2 & 1 & -1 & 2 & -1\cr
                1 & a & -5 &-8 &  a\cr}$$
\item{A)}  (10) Are there any $a \in {\bf R}$ such that
columns 1,2,3 of $A$ form a linearly dependent set?  If so, 
find such an $a$.  If not, say why not.
\item{B)} (10) Are there any $a \in {\bf R}$ such that
the system of equations $Ax = e_1$ (the first standard basis vector
in ${\bf R}^3$) is inconsistent?  If so, for which $a$ is this true?
If not, say why not.
\item{C)} (15)  Let $a = 4$ and find a parametrization for 
${\rm Nul}(A)$.
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\noindent
II. 
\item{A)} (5)  {\it Define:}  The {\it span} of a set $S$ in a 
vector space $V$.  
\item{B)} (10) Given:  if $V$ is a vector space, $S$ is 
a finite subset of $V$ with ${\rm Span}(S) = V$, and
$T$ is any linearly independent subset of $V$,
then $|T| \le |S|$.  Use this fact to show that any
two bases of $V$ must have the same number of elements.
\item{C)} (5) The result of part B
provides the justification for the definition 
of what concept connected to vector spaces? 
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\noindent
III. 
\item{A)} (15) Solve by Cramer's Rule:
$$\eqalign{3x_1\ \ \qquad + x_3 &= 3\cr
           4x_1 + 3x_2 - x_3 &= 2\cr
           \qquad  x_2 + 2x_3 &= 1\cr}$$
\item{B)} (5)  Suppose $\det(A) = 3$, $\det(B) = -2$, and $\det(C) = 4$.
What is $\det(A^t B^{-1} C^2)$?
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\noindent
IV.  All parts of this problem refer to the linear mapping 
$T : P_3({\bf R}) \to {\bf R}^3$ defined by
$$T(p) = \pmatrix{p(-1)\cr p(0) \cr p(1)\cr}$$
\item{A)} (10) Show that the {\it image} of $T$ is a vector subspace
of ${\bf R}^3$ {\it directly} (that is, do not just quote a general theorem).
\item{B)} (10)  Find the matrix of $T$ with respect to the 
basis ${\cal B} = \{1,x,x^2,x^3\}$ in the domain and the standard basis in 
${\bf R}^3$.
\item{C)} (15)  State the change of coordinates formula and use it 
to find the matrix of $T$ with respect to 
the basis ${\cal B}$ in the domain and the basis
$${\cal C} = \left\{\pmatrix{1\cr 1\cr 0\cr}, \pmatrix{1\cr 0\cr 1\cr},
\pmatrix{0\cr 1 \cr 1\cr}\right\}$$
in ${\bf R}^3$.  For full credit, you must show all details of the computation
of any matrix inverses you need.  
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\noindent
V.  (10) Show by mathematical induction: If $A = Q^{-1}BQ$ for some invertible $Q$, 
then for all integers $n\ge 1$, $A^n = Q^{-1} B^n Q$.
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\noindent
VI.  Let $A$ be the symmetric matrix:
$$A = \pmatrix{a & 0 & b\cr
               0 & a & 0\cr
               b & 0 & a\cr}$$
where $a,b$ are real numbers with $b > a > 0$. 
\item{A)} (10) Show that $A$ has eigenvalues $\lambda
= a + b, a, a - b$ and determine the corresponding eigenspaces.    
\item{B)} (10) Find the spectral decomposition of $A$.  
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\noindent
VII.  True - False.  For each true statement give a brief proof
(saying ``this is true by definition'' is {\it not sufficient}!).  For
each false statement give a counterexample or a reason.
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\item{A)} (10) If $A$ is an $n\times n$ matrix
and $AQ = QD$ for some diagonal matrix $D$ and some matrix $Q$,
then $A$ is diagonalizable.  
\item{B)} (10) If $A \in M_{4 \times 5}({\bf R})$ and $\dim {\rm Col}(A) = 2$,
then there are $4$ free variables in the system $Ax = 0$.    
\item{C)} (10) If $\{v_1,v_2,v_3\}$ is a linearly 
independent subset of ${\bf R}^3$ and $V$ is the matrix with columns 
$v_1,v_2,v_3$, then $\det(V) \ne 0$.  
\item{D)} (10) If $A$ is a symmetric matrix and $v_1, v_2, \ldots, v_k$ are
eigenvectors of $A$ with distinct corresponding eigenvalues, 
then $\{v_1, v_2, \ldots, v_k\}$ is an orthogonal set.
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\noindent
VIII.  (The Cayley-Hamilton Theorem.)
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\item{A)}  (10)  Let $A = \pmatrix{a & b \cr c & d\cr}$ be 
any $2\times 2$ matrix.  Show
that $A$ ``satisfies its own characteristic polynomial equation''
in the sense that $A^2 - {\rm Tr}(A) A + \det(A) I_2 = 0_{2\times 2}$
(here $I_2$ is the $2\times 2$ identity matrix and $0_{2\times 2}$
is the $2\times 2$ zero matrix).
\item{B)}  (10) Prove that an $n\times n$ diagonalizable
matrix $A$ satisfies its own characteristic polynomial in the
same sense as in part A:  If 
$$\det(A - tI) = (-1)^n t^n + c_{n-1} t^{n-1} + \cdots + c_1 t + c_0,$$
then 
$$(-1)^n A^n + c_{n-1} A^{n-1} + \cdots + c_1 A + c_0 I_n = 0_{n\times n}.$$
Caution:  It {\it does not} make sense to ``substitute $t = A$'' in
$\det(A - tI)$'' because $t$ is a scalar, not a matrix.
The result of problem V above may be useful.
({\it Comment:} The result is also true without the extra hypothesis of diagonalizability
but it is more difficult to prove then.)
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\noindent
IX.  (Eigenvalues in calculus!)
Let $V = C^{\infty}({\bf R})$ be the vector space of all functions
$f : {\bf R} \to {\bf R}$ that have derivatives of all orders at
all $x \in {\bf R}$.
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\item{A)} (10)  Let $D : V \to V$ be the linear mapping 
defined by $D(f) = f'$.
Show that for all $\lambda \in {\bf R}$, the eigenspace 
${\rm Ker}(D - \lambda I)$ has dimension 1, and find a basis.
\item{B)} (5)  Let $S : V \to V$ be the linear mapping
defined by $S(f) = f''$.  Let $\lambda = -k^2$ for an
arbitrary $k > 0$.  Find two linearly independent functions in 
the eigenspace ${\rm Ker}(S - \lambda I)$.  
\item{C)} (5)  Let $S : V \to V$ be the linear mapping defined by 
$$S(f) = f(1) + f'(1) (x-1) + {f''(1)\over 2}(x-1)^2.$$  
Show that ${\rm Span}\{1,x,x^2\} \subseteq {\rm Ker}(S - I)$.
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\centerline{\it Have a safe, enjoyable, and productive summer!}
\bye

 
