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\centerline{Mathematics 244, section 1 -- Linear Algebra}
\centerline{Review Sheet, Final Exam}
\centerline{April 30, 2007}
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\noindent
{\it General Information}
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The final examination for this course will be given at 
8:30 a.m. on Thursday, May 10 in our regular class room, 
Swords 359.  It will be a comprehensive exam, covering all 
the material we have studied each semester, divided {\it roughly} 
in thirds according to the topics from the three hour exams, plus
a question about material related to the Spectral Theorem 
for real symmetric matrices from the last week of the course. 
Some questions making use of earlier material (especially reduction
to row-reduced echelon form, solving systems of equations, 
etc.) may appear in the context of topics covered later in the 
semester.  Also, there will be one {\it more challenging} problem
that will ask you to put things you have learned together in 
new ways.

As a whole, the exam will be roughly twice the length of one of the three
midterms, but you will have the full three hour period from
8:30 am to 11:30 am to work on it if you need that much time.
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\noindent
{\it Topics to be Covered}
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\noindent
{\it Note:}  This is not quite the order the topics were presented in class; it
shows some logical interconnections, though.
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\item{1)}  Systems of linear equations, row reduction and echelon form, pivot positions,
solution sets of systems of linear equations in parametric form, homogeneous and inhomogeneous systems, the connection 
between the solutions of $Ax = b$ and $Ax = 0$.
\item{2)}  Matrix operations and matrix inverses
\item{3)}  Vector spaces, and the key examples: ${\bf R}^n$, $P_n({\bf R})$,
$M_{m\times n}({\bf R})$, vector spaces of functions, etc.   
\item{4)}  Vector subspaces, including null and column spaces of a matrix
\item{5)}  Linear combinations and spans
\item{6)}  Linear independence and dependence for sets of vectors
\item{7)}  Bases and dimension (know how to find bases for the column and null spaces 
of a given matrix and determine its rank in particular), coordinate systems in a vector space $V$, the 
coordinate vector of a vector with respect to a given basis, change of 
coordinates matrices.
\item{8)} Linear transformations $T: V -> W$ and their matrices, 
the change of coordinates formula for $[T]$.
\item{9)} The determinant of an $n \times n$ matrix, cofactor expansions, 
properties of determinants (in particular, the effect of the various
row operations on determinants, $\det(A) = 0 \Leftrightarrow A$ is not invertible, 
$\det(AB) = \det(A)\det(B)$), Cramer's Rule for solving $Ax = b$ when $\det(A) \ne 0$,
Determinants and areas, volumes.
\item{10)} Eigenvalues and eigenvectors, the characteristic equation.  
\item{11)} Diagonalization and diagonalizability, similarity of 
matrices.
\item{12)} Dot products, orthogonality, orthogonal projections.
\item{13)} The Spectral Theorem for real symmetric matrices. 
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\noindent
{\it Proofs to Know}
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See the review sheets for Exams 1, 2, and 3.  (These
are reposted on the course homepage in case you need
additional copies), and:
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\item{$\bullet$} Show that if $A$ is a real symmetric matrix
then all eigenvalues of $A$ are real, and that
eigenvectors for distinct eigenvalues are orthogonal.  
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\noindent
{\it Suggestions on How to Study}
\bigskip
As I have said before, linear algebra is an {\it extremely
cumulative} subject.  But this can and should actually be working {\it to 
your benefit now}(!)  If you have worked conscientiously
at learning the concepts and techniques we have studied,
things that seemed confusing earlier just might be 
``crystal clear'' now(!)  

To prepare for the final, 
start by reading the above list of topics carefully.
If there are terms there that are unfamiliar or for which
you cannot give the precise definition, {\it learn the definitions
now}.   Review the class notes.  {\it Most of the problems on 
the final will be similar to something we have discussed at
some point this semester, although things might be stated
differently, or appear in a new context.}  As mentioned
above, there may also be a more challenging question
that will require you to put things together in a new way.
Also look back over your graded problem sets and exams.  

If there are problems that you did not get the first time around, 
try them again now.  Then go through the suggested problems from the 
review sheets.  If you have worked these out previously,
it is not necessary to do them all again.  But try a representative
sample ``from scratch'' -- don't just look over your old
solutions and nod your head if it looks familiar.  You
need the practice thinking through the logic of how
the solution is derived again!
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\noindent
{\it Review Session}
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I will be happy to run a review session for the final
exam during study week.  We can discuss a time in class
on Monday, April 30.
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\noindent
{\it Some Sample Exam Questions}
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\noindent
I.  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
II.  Let $V = P_4({\bf R})$, and let 
$W = \{p(x) \in V : p'(1) = 0\ {\rm and}\ p(0) = 3p(2)\}$.
\item{A)}  Show that $W$ is a vector subspace of $V$.
\item{B)}  Find a subset $S \subset W$ such that 
$W = {\rm Span}(S)$.
\item{C)}  Is your set $S$ linearly independent?
Justify your assertion.
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\noindent
III.  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 ${\bf x}$ is a linear combination of the
vectors in ${\rm Span}(S - \{{\bf x}\})$.
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\noindent
IV.  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)}  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
$${\cal B} = \{(1,-1,0,0),(1,1,0,0),(0,0,2,3),(0,0,-1,2)\}$$
in the domain and 
$${\cal C} = \{(0,0,1),(0,-1,1),(-1,1,0)\}$$
in the target.  
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\noindent
V.   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 
$$A = \pmatrix{2&9&0\cr
               1&2&-1\cr
               0&0&5\cr}$$
\item{A)}  Find the eigenvalues of $A$.
\item{B)}  For each eigenvalue $\lambda$, find a basis
of ${\rm Nul}(A - \lambda I)$.
\item{C)}  Is $A$ 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)}  If ${\bf y}, {\bf z}$ are two fixed vectors in 
${\bf R}^n$, then the mapping $T : {\bf R}^n \to {\bf R}^n$
defined by 
$$T({\bf x}) = ({\bf x}\cdot {\bf y}) {\bf y} + 
({\bf x}\cdot {\bf z}) {\bf z}$$
is linear.
\item{B)} 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)} If $A \in M_{2\times 5}({\bf R})$,
then for every $b \in {\bf R}^2$ the general solution of the 
equation $Ax = b$ contains three arbitrary constants.
\item{D)}  If $A$ is an $n\times n$ matrix and $\det(A^2 - 49I) = 0$,
then there exists a nonzero vector $x$ such that $Ax = 7x$.  
\item{E)}  If $A,B \in M_{n\times n}({\bf R})$, and
$B = Q^{-1}AQ$ for some invertible $n\times n$ matrix $Q$, then
$\det(B) = \det(A)$.  
\item{F)}  If $A \in M_{6 \times 9}({\bf R})$ then  
$\dim {\rm Col}(A)$ can equal 8.
\item{G)}  If $A \in M_{6 \times 9}({\bf R})$ then  
$\dim {\rm Nul}(A)$ is at most 6.
\item{H)}  If $A \in M_{6 \times 9}({\bf R})$ and  
$\dim {\rm Nul}(A) = 4$, then $\dim {\rm Col}(A) = 5$.
\item{I)}  If $U$ is a $7\times 3$ matrix whose columns $\{u_1,u_2,u_3\}$
form an orthonormal set in ${\bf R}^7$, then $u_1, u_2, u_3$ are 
eigenvectors of the $7\times 7$ matrix $UU^t$.
\item{J)}  If $A$ is a $3\times 3$ matrix with eigenvalues $\lambda = 2,5,-7$
then Cramer's Rule can be used to solve $Ax = b$ for all
$b in {\bf R}^3$. 
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\noindent
VIII. ({\it more challenging})
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\item{A)}  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)}  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)}  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
IX.  ({\it more challenging})
\item{A)} 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)} Suppose that $A$ is symmetric with 
eigenvalues $\lambda_1, \ldots, \lambda_n$, 
and corresponding eigenvectors ${\bf x}_1,\ldots, {\bf x}_n$ forming an orthonormal
basis of ${\bf R}^n$.  Let $P_1, \ldots, P_n$
be the matrices of the orthogonal projections onto the lines spanned
by the eigenvectors of $A$.  Explain why
$$A^2 = \lambda_1^2 P_1 + \cdots + \lambda_n^2 P_n,$$
using the Spectral Theorem.
\item{C)} Verify that the formula in part B is true in the 
case of the $2\times 2$ symmetric matrix 
$A = \pmatrix{4&1\cr 1&4}$.
\bye

