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\centerline{Mathematics 244, section 1 -- Linear Algebra}
\centerline{Final Examination}
\centerline{May 10, 2004}
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\noindent
{\it Directions}
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Do problems I - VII, and either VIII or VIII' in the blue exam book.
If you wish, you may submit solutions for {\it both} VIII and VIII' 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 $V = P_4({\bf R})$, and let 
$W = \{p(x) \in V : p'(1) = 0\ {\rm and}\ p(0) = 3p(2)\}$.
\item{A)} (10) Show that $W$ is a vector subspace of $V$.
\item{B)} (10) Find a subset $S \subset W$ such that 
$W = {\rm Span}(S)$.
\item{C)} (10) Is your set $S$ linearly independent?
Justify your assertion.
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\noindent
II.  (20) Let $V$ be a vector space.  Show that a set $S \subset V$
is linearly dependent if and only if there is some
${\bf x} \in S$ such that ${\rm Span}(S - \{{\bf x}\}) = {\rm Span}(S)$.
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\noindent
III.  All parts of this problem refer to the linear mapping 
$T : {\bf R}^4 \to {\bf R}^3$ defined by
$$T(x_1,x_2,x_3,x_4) = (4x_1 - 3x_2 + 2x_3 - x_4, x_2 + x_3, x_1 - x_4)$$
\item{A)} (10)  Find the matrix of $T$ with respect to the standard
bases in the domain and target.
\item{B)} (15) Find the matrix of $T$ with respect to the basis
$$\alpha' = \{(1,-1,0,0),(1,1,0,0),(0,0,2,3),(0,0,-1,2)\}$$
in the domain and 
$\beta' = \{(0,0,1),(0,-1,1),(-1,1,0)\}$
in the target.  (You may use any applicable method.)
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\noindent
IV. (20) The following matrix is the augmented matrix for a system
of 3 linear equations in 4 unknowns.  For which value(s) of $a \in {\bf R}$
does the system have a solution?  Find all solutions for those $a$.
$$\pmatrix{-1& 2& 0 &1& |&2\cr
            3& 1& 3 &0& |&0\cr
            1&12& 6 &5& |&a\cr}$$
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\noindent
V.  (20) Show by mathematical induction that the determinant
of an $n\times n$ upper-triangular matrix $A = (a_{ij})$ is 
the product of the diagonal entries:  
$\det(A) = a_{11}a_{22} \cdots a_{nn}$.
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\noindent
VI.  Let $T : {\bf R}^3 \to {\bf R}^3$ be the linear mapping whose
matrix with respect to the standard basis is
$$M = \pmatrix{2&9&0\cr
               1&2&-1\cr
               0&0&5\cr}$$
\item{A)} (10) Find the eigenvalues of $T$.
\item{B)} (15) For each eigenvalue $\lambda$, find a basis
of $E_\lambda = {\rm Ker}(T - \lambda I)$.
\item{C)} (10)  Is $T$ diagonalizable?  Explain.
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\noindent
VII.  True - False.  For each true statement give a brief proof.  For
each false statement give a counterexample or a reason.
\item{A)} (10)  If ${\bf y}, {\bf z}$ are two vectors in 
${\bf R}^n$, then the mapping $T : {\bf R}^n \to {\bf R}^n$
defined by 
$$T({\bf x}) = \langle {\bf x},{\bf y}\rangle {\bf y} + 
\langle {\bf x},{\bf z}\rangle {\bf z}$$
is linear.
\item{B)} (10)  No matter what the entries $a,b,c \in {\bf R}$ are, 
all the roots of the characteristic polynomial of the 
matrix $A = \pmatrix{a&b \cr b&c}$ are real. 
\item{C)} (10)  If $T : V \to W$ is linear, $\dim(V) = 5$, and $\dim(W) = 2$,
then for every ${\bf y} \in W$ the general solution of the 
equation $T({\bf x}) = {\bf y}$ contains three arbitrary constants.
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\noindent
VIII. 
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\item{A)}  (10) Show that there exists a basis $\{p_1(x), p_2(x), p_3(x)\}$
for $V = P_2({\bf R})$ satisfying 
$$\matrix{p_1(0) = 1, &p_1(1) = 0, &p_1(2) = 0\cr
          p_2(0) = 0, &p_2(1) = 1, &p_2(2) = 0\cr
          p_3(0) = 0, &p_3(1) = 0, &p_3(2) = 1\cr}$$
Be sure to say how you know your polynomials form a basis for $V$.
\item{B)}  (5)  Suppose you knew that the polynomial
$q(x)$ satisfied $q(0) = 8$, $q(1) = -3$ and $q(2) = 4$.
What would the expansion of $q(x)$ in terms of the 
basis $\{p_1(x),p_2(x),p_3(x)\}$ be?
\item{C)}  (5)  Generalize what you did in part A to show that
given any three distinct real numbers $a,b,c$, there exists
a basis $\{p_1(x), p_2(x), p_3(x)\}$ of $V = P_2({\bf R})$ satisfying 
$$\matrix{p_1(a) = 1, &p_1(b) = 0, &p_1(c) = 0\cr
          p_2(a) = 0, &p_2(b) = 1, &p_2(c) = 0\cr
          p_3(a) = 0, &p_3(b) = 0, &p_3(c) = 1\cr}$$
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\noindent
VIII'. 
\item{A)} (5) Show that if ${\bf x}$ is an eigenvector of 
a matrix $A$ with 
eigenvalue $\lambda$, then ${\bf x}$ is also an eigenvector of $A^2$.
\item{B)} (5) Suppose that $A$ is symmetric with distinct
eigenvalues $\lambda_1, \ldots, \lambda_k$.  Let $P_1,\ldots,P_k$
be the matrices of the orthogonal projections onto the eigenspaces
$E_{\lambda_1}, \ldots, E_{\lambda_k}$ of $A$ (with respect
to the standard basis).  Explain why
$$A^2 = \lambda_1^2 P_1 + \cdots + \lambda_k^2 P_k,$$
using the Spectral Theorem.
\item{C)} (10) Verify that the formula in part B is true in the 
case of the $2\times 2$ symmetric matrix 
$A = \pmatrix{4&1\cr 1&4}$.
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          \centerline{\it Have a safe, enjoyable, and productive summer!}
          \bye

 
