\begin{problem}{MATH 294}{SPRING 1987}{PRELIM 3}{9}{}
\prob{For problems (a) - (c) use the bases $B$ and $B^{\prime}$
below:} \probnn{$B = \left\{ \left( \begin{array}{c}
  1 \\
  0
\end{array} \right), \left( \begin{array}{c}
  1 \\
  -1
\end{array} \right) \right\}$ and $B^{\prime} = \left\{ \left( \begin{array}{c}
  1 \\
  1
\end{array} \right), \left( \begin{array}{c}
  0 \\
  1
\end{array} \right) \right\}$.} \probpart{Given that $[\vec{v}]_B=\left( \begin{array}{c}
  2 \\
  3
\end{array} \right)$ what is $[\vec{v}]_{B^{\prime}}$?}
\probpart{Using the standard relation between $\Re^2$ and points
on the plane make a sketch with the point $\vec{v}$ clearly
marked.  Also mark the point $\vec{w}$, where $[\vec{w}]_B =
\left( \begin{array}{c}
  0 \\
  -1
\end{array} \right)$. } \probpart{Draw the line defined by the points $\vec{v}$ and $\vec{w}$.  Do the points on this line represent a subspace of $\Re^2$?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1987}{FINAL}{9}{}
\prob{A general vector $\vec{v}$ in $\Re^2$ is $\vec{v} = b_1
\vec{v}_1 + b_2 \vec{v}_2 = b_1^{\prime} \vec{v}_1^{\, \prime} +
b_2^{\prime} \vec{v}_2^{\,\prime}$. where} \probnn{$\vec{v}_1 =
\left( \begin{array}{c}
  1 \\
  0
\end{array} \right), \vec{v}_2 =
\left( \begin{array}{c}
  1 \\
  1
\end{array} \right), \vec{v}_1^{\, \prime} =
\left( \begin{array}{c}
  1 \\
  0
\end{array} \right), \vec{v}_2^{\, \prime} =
\left( \begin{array}{c}
  0 \\
  1
\end{array} \right)$.} \probnn{Find a matrix $_{B^{\prime}}[I]_{B}$ so that $\left( \begin{array}{c}
  b_1^{\prime} \\
  b_2^{\prime}
\end{array} \right) = _{B^{\prime}}[I]_{B} \left( \begin{array}{c}
  b_1 \\
  b_2
\end{array} \right)$ for all vectors $\vec{v}$ in $\Re^2$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING 1993}{FINAL}{5}{}
\probb{Determine the matrix $H_{E,E}$ which represents reflection
of vectors in $\Re^2$ about the y-axis in the standard basis $E =
 \left\{ \left( \begin{array}{c}
  1 \\
  0
\end{array} \right), \left( \begin{array}{c}
  0 \\
  1
\end{array} \right) \right\}$. Verify your answer by evaluating the expression $$ H_{E,E} \left( \begin{array}{c}
  x \\
  y
\end{array} \right). $$ }
\probpart{Now consider a basis $B$ which is obtained by rotating
each vector of the standard basis by 90 degrees in a
counterclockwise direction.  Find the change-of-basis matrices
$(B:E)$ and $(E:B)$.} \probpart{Find $H_{B,B}$ from the formula
$H_{B,B} = (E:B)H_{E,E}(B:E)$.} \probpart{It is claimed that
$H_{B,B}$ is equal to the matrix $H_{E,E}$ which represents a
reflection about the x-axis in the standard basis.  Do you agree?
Give geometrical reasons for your answer by drawing a suitable
picture.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{FALL 1995}{PRELIM 3}{2}{}
\prob{Consider the vector space $\Re^3$ and the three bases:}
\probpartnn{the standard basis $E = \left\{ \left( \begin{array}{c}
  1 \\
  0 \\
  0
\end{array} \right), \left( \begin{array}{c}
  0 \\
  1 \\
  0
\end{array} \right), \left( \begin{array}{c}
  0 \\
  0 \\
  1
\end{array} \right)
\right\}$,} \probpartnn{the basis $B = \left\{ \left(
\begin{array}{c}
  1 \\
  0 \\
  -1
\end{array} \right), \left( \begin{array}{c}
  0 \\
  1 \\
  -1
\end{array} \right), \left( \begin{array}{c}
  1 \\
  2 \\
  3
\end{array} \right)
\right\}$, and} \probpartnn{the basis $C = \left\{ \left(
\begin{array}{c}
  1 \\
  0 \\
  0
\end{array} \right), \left( \begin{array}{c}
  1 \\
  1 \\
  0
\end{array} \right), \left( \begin{array}{c}
  1 \\
  1 \\
  1
\end{array} \right)
\right\}$.} \probpart{Given the $E$ coordinates of a vector
$\vec{x}, [\vec{x}]_E = \left( \begin{array}{c}
  1 \\
  2 \\
  3
\end{array} \right),$ find $[x]_C$.} \probpart{Given the $B$ coordinates of a vector
$\vec{y}, [\vec{y}]_B = \left( \begin{array}{c}
  1 \\
  0 \\
  -1
\end{array} \right),$ find the coefficients $y_j$ in $\vec{y} = \vec{y}_1 \vec{e}_1 + \vec{y}_2 \vec{e}_2 + \vec{y}_3
\vec{e}_3$.} \probpart{Find the change-of-coordinates matrix $_C P
_B$ whose columns consist of the $C$ coordinate vectors of the
basis vectors of $B$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING 1996}{PRELIM 3}{8}{}
\prob{Let $$ B = \left\{ \left[ \begin{array}{c}
  -1 \\
  8
\end{array} \right] , \left[ \begin{array}{c}
  1 \\
  -5
\end{array} \right] \right\} , C = \left\{ \left[ \begin{array}{c}
  1 \\
  4
\end{array} \right] , \left[ \begin{array}{c}
  1 \\
  1
\end{array} \right] \right\}. $$} \probpart{Find the change of coordinate matrix from $B$ to
$C$.} \probpart{Find the change of coordinate matrix from $C$ to
$B$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING 1996}{FINAL}{8}{}
\prob{Let $$ B = \left\{ \left[ \begin{array}{c}
  1 \\
  1
\end{array} \right] , \left[ \begin{array}{c}
  2 \\
  0
\end{array} \right] \right\} , C = \left\{ \left[ \begin{array}{c}
  2 \\
  2
\end{array} \right] , \left[ \begin{array}{c}
  2 \\
  -2
\end{array} \right] \right\}. $$} \probnn{Then the change of coordinates matrix from coordinates with respect to the basis $C$ to coordinates with respect to the basis $B$
is} \probpart{$\left( \begin{array}{cc}
  2 & -2 \\
  0 & 2
\end{array} \right)$}
\probpart{$\left( \begin{array}{cc}
  -4 & 4 \\
  0 & -4
\end{array} \right)$}
\probpart{$\left( \begin{array}{cc}
  0 & \frac{1}{2} \\
  \frac{1}{2} & \frac{1}{2}
\end{array} \right)$}
\probpart{$\left( \begin{array}{cc}
  0 & 4 \\
  4 & 4
\end{array} \right)$}
\probpart{none of the above}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1995}{PRELIM 3}{8}{}
\prob{You are given a vector space $V$ with an inner product $<,>$
and an orthogonal basis $B = \{ \vec{b}_1 , \vec{b}_2 , \vec{b}_3
, \vec{b}_4 , \vec{b}_5$ for $V$ for which $\left| \left|
\vec{b}_i \right| \right| = 2, i = 1, \ldots, 5$. Suppose that
$\vec{v}$ is in $V$ and $$ \left< \vec{v},\vec{b}_1\right> =
\left< \vec{v},\vec{b}_2\right> = 0$$  $$ \left<
\vec{v},\vec{b}_4\right> = 3, \left< \vec{v},\vec{b}_4\right> = 4,
\left< \vec{v},\vec{b}_5\right> = 5$$ Find the coordinates of
$\vec{v}$ with respect to the basis $B$ i.e. find
$c_1,c_2,c_3,c_4,c_5$ such that $$ \vec{v} = c_1 \vec{b}_1 + c_2
\vec{b}_2 + c_3 \vec{b}_3 + c_4 \vec{b}_4 + c_5 \vec{b}_5$$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1998}{PRELIM 3}{3}{}
\prob{Let $T:\wp_1 \rightarrow \wp_3$ be defined by \[ T\left[
p(t)\right] =t^2 p(t)\]and take \[ B = \left\{ 1,1+t \right\} \]
to be the basis of $\wp_3$.} \probpart{Find the matrix of $T$
relative to the bases $B$ and $C$.} \probpart{Use this matrix to
find $T[2+t]$.} \probpart{Let $E = \left\{1,t\right\}$ be the
standard basis for $\wp_1$. Let $\left[ \vec{x} \right]_B$ be the
coordinate vector of $\vec{x}$ in $\wp_1$ relative to the basis
$B$, and let $\left[ \vec{x} \right]_E$ be the coordinate vector
of $\vec{x}$ relative to the basis $E.$ What is the change of
coordinate matrix $P.$ such that \[ P\left[ \vec{x} \right] _B =
\left[ \vec{x} \right]_E. \] } \probnn{[Note: The result of part
c) does not depend on the results of parts a) or b)] }
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1998}{Final}{4}{}
\prob{In $P^2$, Find the change-of-coordinate matrix from the
basis $$B = \left\{ 1-2t + t^2 , 3-5t,2t+3t^2\right\}$$ to the
standard basis \[E = \{1,t,t^2\}. \]  Then write $t^2$ as a
linear combination of the polynomials in $B$, i.e. give the
coordinates of $t^2$ with respect to the basis $B$. }
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{Fall 1998}{PRELIM 2}{3}{}
\prob{Besides the standard basis $\varepsilon$ here are two bases
for $\Re^2$:
\[ B =
        \left\{
            \underbrace
            {
                \left[
                    \begin{array}{c}
                      1 \\
                      1 \\
                        \end{array}
                \right],
            }_
            {
                b_1
            }
            \underbrace
            {
                \left[
                    \begin{array}{c}
                        -1 \\
                        1 \\
                    \end{array}
                \right]
            }_
            {
                b_2
            }
            \right\}
        , C =
        \left\{
            \underbrace
            {
                \left[
                    \begin{array}{c}
                      2 \\
                      4 \\
                        \end{array}
                \right],
            }_{c_1}
            \underbrace
            {
                \left[
                    \begin{array}{c}
                        -4 \\
                        4 \\
                    \end{array}
                \right]
            }_{c_2}
            \right\}
\]
} \probpart{What vectors $\vec{x}$ are represented by
    $
        \left[
            \vec{x}
        \right]_B
        =
        \left[
            \begin{array}{c}
                2 \\
                14 \\
            \end{array}
        \right]
    $ and
    $
        \left[
            \vec{x}
        \right]_C
        =
        \left[
            \begin{array}{c}
                2 \\
                4 \\
            \end{array}
        \right]
    $?
} \probpart{ Find a single tidy formula to find the components
    $
        \left[
            \begin{array}{c}
                d \\
                e \\
            \end{array}
        \right]
    $
    of a vector $\vec{x}$ in the basis $B$ if you are given the
    components
    $
        \left[
            \begin{array}{c}
                f \\
                g \\
            \end{array}
        \right]
    $
    of $\vec{x}$ in the basis $C$.
} \probpart{ A student claims that the desired formula is
    $
        \left[
            \begin{array}{c}
                d \\
                e \\
            \end{array}
        \right]
        =
        \left[
            \begin{array}{cc}
              5 & -2 \\
              5 & 1 \\
            \end{array}
        \right]
        \left[
            \begin{array}{c}
                f \\
                g \\
            \end{array}
        \right].
    $
    Does this formula make the right prediction for the component
    vector $
        \left[\vec{x}\right]_C =
        \left[
            \begin{array}{c}
                f \\
                g \\
            \end{array}
        \right] =
        \left[
            \begin{array}{c}
                2 \\
                4 \\
            \end{array}
        \right]
    $?
}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1998}{Final}{6}{}
\prob{Let $A = \left[
    \begin{array}{cc}
      1 & 1 \\
      1 & 1
    \end{array}
\right].$} \probpart{Find orthogonal eigenvectors $\{ \vec{v}_1,
\vec{v}_2 \}$ of $A$. [Hint: do not go on to parts d-e below until
you have double checked that you have found two orthogonal unit
vectors that are eigenvectors of $A$.]} \probpart{Use the
eigenvectors above to diagonalize $A$.} \probpart{Make a clear
sketch that shows the standard basis vectors
$\{\vec{e}_1,\vec{e}_2\}$ of $\Re^2$ and the eigenvectors
$\{\vec{v}_1,\vec{v}_2\}$ of $A$} \probpart{Give a geometric
interpretation of the change of coordinates matrix, $P$, that
maps coordinates of a vector with respect to the eigen basis to
coordinates with respect to the standard basis.} \probpart{Let
$\vec{b} =
\left[
    \begin{array}{c}
      3 \\
      5
    \end{array}
\right]. $ Using orthogonal projection express $\vec{b}$ in terms
of $\{\vec{v}_1,\vec{v}_2\}$ the eigenvectors of $A$.}
\end{problem}
%----------------------------------
