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\begin{document}





\author{B. Hall and R. Smarandache}

\title{Math 228, Final Exam}

\date{December 15, 2000}

\maketitle







\textbf{Instructions}: There are 17 problems with a total of 150 possible

points. The point-value of each problem is indicated. Show your work clearly

on all problems, except on the true/false problems and where noted.



You have two hours for the exam.



Calculators, notes, and books are prohibited.



This exam is bound by the provisions of the Notre Dame Honor Code.\bigskip



\textbf{Name}: \bigskip



\begin{enumerate}

\item (8 points) If

\[

\det\left[

\begin{array}

[c]{ccc}%

a & b & c\\

m & n & p\\

x & y & z

\end{array}

\right]  =3,

\]

what is the value of the determinant

\[

\det\left[

\begin{array}

[c]{ccc}%

3a & 3b & 3c\\

m+a & n+b & p+c\\

-x & -y & -z

\end{array}

\right]  ?

\]

\pagebreak 



\item (8 points) Suppose that the population $P\left(  t\right)  $ of bacteria

in a certain culture satisfies a differential equation of the form

\[

\frac{dP}{dt}=kP

\]

for some constant $k.$ Suppose that the initial population of bacteria is

10,000 and that the population after one day is 30,000. Determine the constant

$k$ and find the formula for $P\left(  t\right)  .$ How long will it take the

population of bacteria to reach 1,000,000?



\textit{Note}: Your answer will involve logarithms, which you do not have to

evaluate numerically.\pagebreak 



\item (8 points.) Let $A$ be the matrix

\[

\left[

\begin{array}

[c]{rrr}%

2 & 3 & -1\\

4 & -1 & -2\\

-2 & -1 & 1

\end{array}

\right]  .

\]



a) Solve the system

\[

Ax=\left[

\begin{array}

[c]{r}%

-5\\

4\\

1

\end{array}

\right]  .

\]



b) Find a basis for $NS\left(  A\right)  .$



c) Is zero an eigenvalue of $A?$ Why or why not?\pagebreak \pagebreak 



\item (8 points) Find the solution of the equation

\[

y^{\prime\prime}-4y^{\prime}+4y=0

\]

subject to the initial conditions

\begin{align*}

y\left(  0\right)   & =0\\

y^{\prime}\left(  0\right)   & =1.

\end{align*}

\pagebreak 



\item (10 points) a) Compute the inverse of the matrix

\[

A=\left[

\begin{array}

[c]{rrr}%

2 & 1 & 0\\

-4 & -1 & -3\\

4 & 1 & 2

\end{array}

\right]  .

\]

b) Find the L-U decomposition of $A.$ No row interchanges should be necessary.\pagebreak 



\item (8 points) Solve the differential equation

\[

\frac{dy}{dx}=y\left(  x+1\right)

\]

subject to the initial condition

\[

y\left(  1\right)  =2.

\]

\pagebreak 



\item (10 points) a) Let

\begin{align*}

p_{1}\left(  x\right)   & =x^{2}-x+1\\

p_{2}\left(  x\right)   & =x+2\\

p_{3}\left(  x\right)   & =2x^{2}-3x.

\end{align*}

Is the set $S=\left\{  p_{1},p_{2},p_{3}\right\}  $ a basis for $\mathcal{P}%

_{2}?$



b) Let $S=\left\{  \left(  1,0,0\right)  ,\left(  2,0,0\right)  ,\left(

0,1,1\right)  ,\left(  2,2,2\right)  ,\left(  0,3,3\right)  \right\}  .$ Find

a basis for the span of $S.$\pagebreak 



\item (8 points) Find the general solution, in a form involving only real

numbers, of the equation

\[

y^{\prime\prime}+y^{\prime}+y=0.

\]

Describe the behavior of the system as $t$ increases.\pagebreak 



\item (10 points) In each of the following, you are given a vector space $V$

and a subset $S$ of $V.$ In each case, state whether or not $S$ is a subspace

of $V.$ If so, state the dimension of $S.$ (No work is required; credit will

be assigned strictly on your answers.)\medskip



a) $V=\mathbb{R}^{3},$ $S=\left\{  \left(  x,y,z\right)  |\,z=x+y\right\}

\smallskip$



b) $V=\mathbb{R}^{3},$ $S=\left\{  \left(  x,y,z\right)  |\,z=x+y+1\right\}

\smallskip$



c) $V=\mathbb{R}^{3},$ $S=\left\{  \left(  0,t+1,2t+2\right)  |\,t\in

\mathbb{R}\right\}  \smallskip$



d) $V=C\left[  0,2\right]  ,$ $S=\left\{  f\in V|\,f\text{ is a polynomial of

degree at most 2}\right\}  \smallskip$



e) $V=\mathcal{P}_{3},$ $S=\left\{  p\in V\left|  \int_{0}^{1}p\left(

x\right)  \,dx=1\right.  \right\}  \vspace{2.5in}$



\item (8 points) For each of the following, determine whether or not the

statement is always true.\medskip



a) If $Q$ is an orthogonal $n\times n$ matrix, then $Q^{T}$ is orthogonal.\smallskip



b) If $Q$ is an orthogonal $n\times n$ matrix, then $Q^{2}$ is orthogonal.\smallskip



c) If $A$ is an $n\times n$ matrix and $v$ is an eigenvector for $A$ with

eigenvalue $\lambda,$ then $3v$ is an eigenvector for $A$ with eigenvalue

$3\lambda.\smallskip$



d) If $A$ and $B$ are symmetric $n\times n$ matrices, then $AB$ is symmetric.\pagebreak 



\item (8 points) Find the general solution of the equation

\[

\frac{dy}{dt}-\frac{1}{t}y=t^{2}.

\]

\pagebreak 



\item (8 points) What is the largest interval on which the solution of

\begin{align*}

\sqrt{2-t^{2}}\frac{dy}{dt}+e^{t}\,y  & =t^{3}\\

y\left(  0\right)   & =1

\end{align*}

is guaranteed to exist?\pagebreak 



\item (10 points) Determine whether each of the following two differentials is

exact. In each case, solve the equation, using an integrating factor if

necessary.

\[%

\begin{array}

[c]{cccc}%

a) & y\,dx+\left(  x+6y\right)  \,dy & = & 0\\

&  &  & \\

b) & 2\sin y\,\,dx+x\cos y\,\,dy & = & 0

\end{array}

\]

\pagebreak 



\item (8 points) Suppose that $A$ is a $3\times3$ matrix whose characteristic

polynomial is $\left(  \lambda+1\right)  \left(  \lambda+2\right)  \left(

\lambda+5\right)  .$ For each of the following statements, determine whether

or not it is always true.\medskip



a) $A$ is invertible.\smallskip



b) $A$ is diagonalizable.\smallskip



c) There exist 3 linearly independent eigenvectors for $A.\smallskip$



d) If $v_{1}$ is an eigenvector for $A$ with eigenvalue $-1$ and $v_{2}$ is an

eigenvector for $A$ with eigenvalue $-5,$ then $v_{1}$ and $v_{2}$ are orthogonal.\pagebreak 



\item (10 points) Find the general solution of the equation

\[

y^{\prime\prime}-y^{\prime}-2y=t+\sin t.

\]

\pagebreak 



\item (10 points) Find an orthonormal basis for $\mathbb{R}^{3}$ consisting of

eigenvectors for the matrix

\[

A=\left[

\begin{array}

[c]{rrr}%

1 & -2 & -2\\

-2 & 1 & 2\\

-2 & 2 & 1

\end{array}

\right]  .

\]

\textit{Hint}: the characteristic polynomial of $A$ is $\left(  \lambda

+1\right)  ^{2}\left(  \lambda-5\right)  .$\pagebreak 



\item (10 points) Suppose that $A$ is an arbitrary $2\times2$ matrix whose

characteristic polynomial is $\lambda^{2}+5\lambda+4.$ What is the

characteristic polynomial of $A^{2}\,?$ Explain your reasoning.

\end{enumerate}

\end{document}