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{\bf Math 265 Linear Algebra\hfill Sample Final\hfill Fall 2003}
\bigskip

\item{1.}It is given that $A=
\bmatrix\format \r&\quad \r&\quad \r&\quad \r&\quad \r\ \\
 1 & 0 & -2 & 1 & 3\\  -1 & 1 & 5 & -1 & -3\\ 0 & 2 & 6 & 0 & 1\\
1 & 1 & 1 & 1 & 4\endbmatrix$, 
$\text{\bf rref\,}(A)=
\bmatrix\format \r&\quad \r&\quad \r&\quad \r&\quad \r\ \\
 1 & 0 & -2 & 1 & 0\\  0 & 1 & 3 & 0 & 0\\ 0 & 0 & 0 & 0 & 1\\ 
0 & 0 & 0 & 0 & 0\endbmatrix$.
\medskip

\itemitem{(a)} Find the rank of $A$.

\itemitem{(b)} Find the nullity of $A$.

\itemitem{(c)} Find a basis for the column space of $A$.  

\itemitem{(d)} Find a basis for the row space of $A$.

\itemitem{(e)} Find a basis for the null space of $A$.

\itemitem{(f)} Find a basis for the orthogonal complement of the row space of
$A$.

\bigskip

\item{2.}Find an equation relating $a$, $b$ and $c$ so that the linear system

$\qquad \left\{
\alignedat3
2x &\ +\ & 2y &\ + & 3z &=a\\
3x &\ -\ & y &\ +\ & 5z &=b\\
x &\ -\ & 3y &\ +\ & 2z &=c\endalignedat\right.$

is consistent for any values $a$, $b$ and $c$ which satisfy that equation.
\bigskip

\item{3.}Determine the values of $a$ so that the linear system

$\qquad\left\{
\alignedat3
x &\ +\ & y &\ +\ & z &=2\\
2x &\ +\ & 3y &\ +\ & z &=5\\
2x &\ +\ & 3y &\ +\ & (a^2-3)z &=a+1\endalignedat\right.$

has (a) no solution, (b) a unique solution, and (c) infinitely many solutions. 
\bigskip

\item{4.}Find the value of $y$ in the solution of the linear system

$\qquad\left\{
\alignedat3
2x &\ +\ & 123y &\ -\ & z &=1\\
x &\ +\ & 456y &\ +\ & z &=0\\
2x &\ +\ & 789y &\ +\ & z &=1\endalignedat\right.$


if you know that the deteminant of the matrix of the above system is equal to $-420$.


\item{5.}Let $L: \Bbb R^3\to \Bbb R^3$ be the linear transformation such that

$$
L\left(\bmatrix 1\\ 0\\ 0\endbmatrix\right)=\bmatrix \format &\r\ \\
-3\\ 1\\ 2\endbmatrix,\ L=\left(\bmatrix \format &\r\ \\ 0\\ 1\\
0\endbmatrix\right)=
\bmatrix\format &\r\ \\ 2\\ -2\\ 
1\endbmatrix,\ L\left(\bmatrix \format &\r\ \\ 0\\ 0\\
1\endbmatrix\right)=\bmatrix \format &\r\ \\ -1\\ 2\\ 4\endbmatrix.
$$
Find the standard matrix for $L$.
\bigskip

\item{6.}Let $L:\Bbb R^3\to \Bbb R^3$ be the linear transformation such that
$$
L\left(\bmatrix\format &\r\ \\ -2\\ 1\\ -2\endbmatrix\right)=\bmatrix \format &\r\ \\ 
-3\\ 1\\  2\endbmatrix,\ L=\left(\bmatrix \format &\r\ \\ 3\\ 2\\ -1\endbmatrix\right)=
\bmatrix\format &\r\ \\ 2\\ -2\\ 1\endbmatrix,\ L\left(\bmatrix \format &\r\ \\ 
-1\\ -1\\ 1\endbmatrix\right)=\bmatrix \format &\r\ \\ -1\\ 2\\ 4\endbmatrix.
$$
Find  $L\left(\bmatrix\format &\r\ \\ 1\\ 1\\ -1\endbmatrix\right)$.
\bigskip

\item{7.}Find the standard matrix of the linear transformation $L$ defined by

$\qquad L\left(\bmatrix\format &\r\ \\ x\\ y\\ z\endbmatrix\right)=
 \bmatrix \format &\c\ \\ 2x -z\\ x  + 2y  + z\\ 3x - y \endbmatrix$
\bigskip

\item{8.} If $A$ is a $5\times 5$ matrix and $\det A=3$ find
$\det (A^2\cdot ($adj$A)^2)$.
\bigskip


\item{9.}Find a basis for  $S=\{ t^2+1,\ 
t^2+2t,\  3t^2+t-1\}$. Does  $ 6t^2 -1$ belong to  span$(S)$?
\bigskip

\item{10.}For what values of $d$ are the vectors $[1, \ 3, \ d]$, $[1, \ 1, \ 0]$ and
$[0, \ 1, \ 1]$ linearly independent?
\bigskip

\item{11.}Compute adj$(A)$ and $A^{-1}$ for $A=\bmatrix \format \r&\quad \r&\quad \r\ \\ 
2 & 0 & 1\\  -2 & 1 & 3\\  2 & -1 & 1\endbmatrix$.
\bigskip

\item{12.}Find ALL diagonal matrices similar to $A=\bmatrix \format \r&\quad \r\ \\ 
1 & 1\\  -5 & 7\endbmatrix$.
\bigskip

\item{13.}Let us consider a triangle $\Delta(P,Q,R)$ in ${\Bbb R}^3$ whose
vertices are as follows: 
$$P=(-2\sqrt{3},1,-2), \ Q=(3\sqrt{3},-3,1),\  R=(-2\sqrt{3},7,6)$$
\item{(a)} Find length of all sides of $\Delta(P,Q,R)$.
\item{(b)} Find all angles of $\Delta(P,Q,R)$.


\item{14.}Use the Gram-Schmidt process to construct an orthonormal basis for the
subspace $W$ of $\Bbb R^4$ spanned by
$$
\left \{
\bmatrix 1 \\1 \\ 0\\ 0\endbmatrix, \
\bmatrix 2\\ -1\\ 0\\ 1\endbmatrix,\
\bmatrix 3\\ -3\\ 0\\ -2\endbmatrix, \
\bmatrix 1\\ -2\\ 0\\ -3\endbmatrix \right \}.
$$

\bigskip


\item{15.}Let $W$ be the subspace of $\Bbb R^3$ spanned by the vector $w=[1 \ 1\ -1]$.

\itemitem{(a)}Find a basis for the orthogonal complement $W^\perp$ of $W$.

\itemitem{(b)}Find an orthonormal basis for $W^\perp$.
\bigskip

\item{16.} Let $W$ be the subspace of $\Bbb R^4$ spanned by the sset of vectors
$
\left \{
\bmatrix 1 \\2 \\ 0\\ -2\endbmatrix, \
\bmatrix 0\\ 1\\ 2\\ 1\endbmatrix,\
\bmatrix 4\\ -1\\ 0\\ 1\endbmatrix \right \},
$ and $\bold v=\bmatrix 1 \\-1 \\ 0\\ 4\endbmatrix$.
 (a) Find the projection of $\bold v$
onto $W$. (b) Find the distance from $\bold v$ to $W$.
\bigskip

\item{17.}For each of the following sets of vectors, determine if it is a vector
(sub)space:

\itemitem{(a)}The set of all vectors in $\Bbb R^4$ with the property\newline
$2x_1+x_2-3x_3+x_4=0$;\hfill Yes\qquad No
\bigskip

\itemitem{(b)}The set of all vectors in $\Bbb R^4$ with the property 
$x_1^3=x_2^3$, $x_3=x_4$;\hfill Yes\qquad No
\bigskip

\itemitem{(c)}The set of all vectors in $\Bbb R^3$, which have the form
$(0, a-b+c, 3b+c)$\newline where $a$, $b$ and $c$ are arbitrary real
numbers;\hfill Yes\qquad No
\bigskip

\itemitem{(d)}The set of all polynomials $P$ in the space of all polynomials of
degree\newline
at most $7$, with the property $P(2)=0$;\hfill Yes\qquad No
\bigskip

\itemitem{(e)}The set of all nonsingular matrices in the space of $3\times 3$
matrices;\hfill Yes\qquad No
\bigskip



\item{18.}Determine if each of the following sets of vectors is linearly
independent or linearly dependent:

\itemitem{(a)} $\left\{\bmatrix 2\\ 1\\ 1\\ 3\endbmatrix,
\bmatrix 3\\ 0\\ 1\\ 1\endbmatrix, \bmatrix 4\\ 1\\ 1\\ 2\endbmatrix\right\}$;
\hfill Independent\qquad Dependent

\itemitem{(b)} $\left\{\bmatrix 1\\ 0\\ 0\endbmatrix, \bmatrix 1\\ 1\\ 
1\endbmatrix, \bmatrix 0\\ 4\\ 5\endbmatrix\right\}$;\hfill Independent\qquad Dependent

\itemitem{(c)} $\left\{\bmatrix 1\\ 0\\ 0\endbmatrix, \bmatrix 1\\ 1\\ 
1\endbmatrix, \bmatrix 0\\ 4\\ 4\endbmatrix\right\}$;\hfill Independent\qquad Dependent

\itemitem{(d)} $\left\{\bmatrix 1\\ 2\\ 5\endbmatrix, \bmatrix 2\\ 1\\ 
7\endbmatrix, \bmatrix 0\\ 1\\ 1\endbmatrix, \bmatrix\format &\r \\ -3\\ 2\\ -1\endbmatrix\right\}$;\hfill Independent\qquad Dependent

\itemitem{(e)} $\left\{\bmatrix\format &\r \\ 1\\ 2\\ -1\endbmatrix, \bmatrix
1\\ 0\\ 1\endbmatrix\right\}$.\hfill Independent\qquad Dependent

\item{19.}We have a $3\times 3$ matrix $A=\bmatrix a & 1 & 2\\  b & 3 & 4\\
c & 5 & 6\endbmatrix$ with $\det(A)=3$.  Compute the determinant of the following matrix

\itemitem{(a)} $\bmatrix a-2 & 1 & 2\\ b-4 & 3 & 4\\  c-6 & 5 & 6\endbmatrix$
\smallskip
\itemitem{(b)} $\bmatrix\format &\r\quad &\r\quad &\r\ \\  7a & 7 & 14\\
b & 3 & 4\\  c & 5 & 6\endbmatrix$
\smallskip
\itemitem{(c)} $\bmatrix a & 1 & 1\\  b & 3 & 3\\  c & 5 & 5\endbmatrix$
\smallskip
\itemitem{(d)} $A^T$

\itemitem{(e)} $(5A)^{-1}$
\bigskip


\item{20.} Let $A=\bmatrix 3 & -5\\  1 & -3\endbmatrix$.

\itemitem{(a)}Find all eigenvalues of $A$.

\itemitem{(b)} For each eigenvalue above, find an eigenvector of $A$ associated
to it.

\itemitem{(c)}Find a diagonal matrix $D$ and a nonsingular matrix $P$ such that
$P^{-1}AP=D$.

\itemitem{(d)}Find $A^{47}$.
\bigskip

\item{21.} Let $A=\bmatrix 2 & 1 & 1\\  1 & 2 & 1\\  1 & 1 & 2\endbmatrix$.

\itemitem{(a)}Find all eigenvalues of $A$.

\itemitem{(b)}For each eigenvalue as above, find an eigenvector of $A$ associated to it.

\itemitem{(c)}Find a diagonal matrix $D$ and an orthogonal matrix $P$ such
that $P^TAP=D$.
\bigskip

\item{22.}Determine an invertible matrix $\hat A$ and a vector $\hat b$ such
that the solution to $\hat A\hat X=\hat b$ is the least squares solution to
$AX=b$, where
$$
A=\bmatrix 1 & 0\\  0 & 1\\ 1 & 1\\ 1 & 0\\ 1 & 0\endbmatrix \text{ and }
b=\bmatrix 1\\ 1\\ 1\\ 1\\ 1\endbmatrix.
$$
\item{23.}Find the least squares fit line for the points
$$
(1,2),(2,1),(3,3), (4,3).
$$

\bigskip

\item{24.} Consider the homogeneous linear system of differential equations
$
{\bold x}'(t)=A{\bold x}$, where
$\qquad A=\bmatrix\format &\r\quad &\r\\ 2 & -1\\  8 & 8\endbmatrix$.

\itemitem{(a)}Find all eigenvalues of $A$.

\itemitem{(b)} Find the general solution to the linear system of differential
equations.

\itemitem{(c)} Solve the initial value problem
for  the given conditions 
$x_1(0)=3$ and $x_2(0)=-10$.
\bigskip



\bye


%%%%%%%%%%%%%%%
\item{  .} Let $A=\bmatrix 2 & 1 & 1\\  -1 & 2 &-1\\  -1 & 1 & 2\endbmatrix$.

\itemitem{(a)} Compute det$A$.
\itemitem{(b)} Compute adj$A$.
\itemitem{(C)} Compute $A^-1$.

\item{9.} If $adj  \ A=\bmatrix\format\r&\quad\r&\quad\r\\
28 & -21 & 11\\ 24 & 12 & -12\\ -8&6&14\endbmatrix$ and $\det A > 0$ find
$A^{-1}$.
\bigskip


\item{31.} Consider the homogeneous linear system of differential equations
$
{\bold x}'(t)=A{\bold x}$, where
$
A=\bmatrix 1 & -2\\  2 & -1\endbmatrix.
$

\itemitem{(a)}Find all eigenvalues of $A$.

\itemitem{(b)}Find the real-valued 
general solution to the linear system of differential equations.

\itemitem{(c)}Solve the initial value problem
for the  given  conditions 
$ x_1(0)=5$ and  $\ x_2(0)=1$.
\bigskip


\item{32.} Consider the homogeneous linear system of differential equations
$
{\bold x}'(t)=A{\bold x}$, where
$
A=\bmatrix\format &\r\quad &\r\quad &\r\\  1 & 2 & -1\\  1 & 0 & 1\\  4 & -4 & 5\endbmatrix.
$

\itemitem{(a)}Find all eigenvalues of $A$.

\itemitem{(b)}Find the general solution to the linear system of differential
equations.

\itemitem{(c)} Solve the initial value problem
for the given conditions
$$x_1(0) =-2,\ x_2(0)=1,\ x_3(0)=2$$.



\item  Consider the following linear system.
\[
\begin{array}{ccccccc}
x &+ &ay &+ &z &= &b+c \\
2x &+ &by &+ &z &= &a+c \\
3x &+ &cy &+ &z &= &a+b \\
\end{array}
\]where
\[ \det \left[\begin{array}{lll}1 & a & 1 \\ 2 &b&1 \\ 3 & c&1  \end{array}  \right] \neq 0 .\]

Using Cramer's rule we find that
\begin{enumerate}
\item $x=0$
\item $x=a$
\item $x=b$
\item $x=1$
\item $x=a+b+1$
\end{enumerate}
