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 {\bf Name~}\underline{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}\hfill
 {Class Time~}\underline{~~~~~~~~}\newline


\begin{center}
\bf Math 2270 Extra Credit Problems \\
\bf Chapter 3 \\
{\small\bf December 2011}
\end{center}

These problems were created for Bretscher's textbook, but apply for Strang's book, except for the division by chapter.
To find the background for a problem, consult Bretscher's textbook, which can be checked out from the math library or the LCB Math Center.

{\bf Due date}: See the internet due dates.
Records are locked on that date and only corrected,
never appended.

{\bf Submitted work}. Please submit one stapled package.
Kindly label problems \fbox{\bf Extra Credit}. Label each problem with
its corresponding problem number. You may attach this printed sheet to
simplify your work.

\Prob{XC3.1-12}{(Image and kernel)}
For the matrix $C$ below, display a frame sequence from $C$ to $\rref(C)$.
Write the image of $C$
as the span of the pivot columns of $C$. Write the kernel of $C$
as the list of partial derivatives $\partial\.x/\partial c_1$, etc,
where $\.x$ is the vector general solution to $C\.x=\.0$.

$$
 C=\left(\begin{array}{rrrrr}
    1 & -1 & -1 & 1 & 1 \\
   -1 &  1 &  0 & 3 & 1 \\
    1 & -1 & -2 & 0 & 4 \\
    1 & -1 & -3 & 4 & 6
   \end{array}\right)
$$

\Prob{XC3.1-22}{(Geometry of a linear transformation)}
Give an example of a $3\times 3$ matrix $A$ such that $T(\.x)=A\.x$
has image equal to the plane through the three points $(0,0,0)$,
$(0,1,1)$, $(1,1,0)$.


\Prob{XC3.1-38}{(Image and kernel)}
Express the image of the matrix $A$ as the kernel of a matrix $B$.
The matrix $B$ can be a different size than $A$.
$$
 A=\left(\begin{array}{rrrr}
    1 & -1 & -1 & 1  \\
   -1 &  1 &  0 & 3  \\
    1 & -1 & -2 & 0  \\
    1 & -1 & -3 & 4
   \end{array}\right)
$$

\Prob{XC3.1-50}{(Kernel and image)}
Let $A$ be an $n\times n$ matrix and $B=\rref(A)$. Do $A$ and $B$ have
the same kernels and images? Prove each assertion or give a counterexample.

\Prob{XC3.2-18}{(Redundant vectors)}
Identify the redundant vectors in the list, by application of the pivot
theorem.
$$
\colvector{1 \\ 1\\ 0 \\ 1},\quad
\colvector{-1 \\ 2\\ 0 \\ 0},\quad
\colvector{0 \\ 12\\ 0 \\ 4},\quad
\colvector{0 \\ 3\\ 0 \\ 1}.
$$

\Prob{XC3.2-22}{(Pivot columns)}
Express the non-pivot columns of $A$ as linear combinations of the pivot
columns of $A$.
$$
 A=\left(\begin{array}{rrrr}
    1 & -1 &  0 & 0  \\
   -1 &  1 &  0 & 0  \\
    1 &  1 &  2 & 3  \\
    1 &  2 &  3 & 4
   \end{array}\right)
$$


\Prob{XC3.2-46}{(Basis for $\ker(A)$)}
Find a basis for the kernel of $A$, using frame sequence methods.

$$
 A=\left(\begin{array}{rrrr}
    0 &  1 &  0 & 0  \\
    0 & -1 &  0 & 0  \\
    0 &  0 &  2 & 3  \\
    0 & -1 &  2 & 3
   \end{array}\right)
$$


\Prob{XC3.2-48}{(Independence with symbols)}
Determine all values of the symbols $a$ and $b$ such that the
following vectors are linearly independent.
$$
\colvector{a \\ 0  \\ 0 \\ 0},\quad
\colvector{a \\ 2b \\ 0 \\ 2b},\quad
\colvector{1 \\ 3b \\ 0 \\ a-b}.
$$


\Prob{XC3.3-10}{(Basis for $\ker(A)$ and $\image(A)$)}
Find redundant columns by inspection and then find a basis for
$\image(A)$ and $\ker(A)$.
$$
 A=\left(\begin{array}{rrr}
    1 &  0 & 0  \\
   -1 &  0 & 0  \\
    2 &  2 & 2
   \end{array}\right)
$$


\Prob{XC3.3-24}{(Basis for $\ker(A)$ and $\image(A)$)}
Find $\rref(A)$ and then a basis for $\ker(A)$ and $\image(A)$.
$$
 A=\left(\begin{array}{rrrr}
    0 &  0 &  0 & 0  \\
    0 &  0 &  0 & 0  \\
    0 &  0 &  2 & 3  \\
    0 &  0 &  1 & 3
   \end{array}\right)
$$

\Prob{XC3.3-52}{(Row space basis)}
Find a basis for the row space of $A$ consisting of columns of $A^T$.
$$
 A=\left(\begin{array}{rrrr}
    0 &  0 &  0 & 0  \\
    1 &  0 &  1 & 0  \\
    3 &  3 &  1 & 3  \\
    5 &  3 &  1 & 3
   \end{array}\right)
$$

\Prob{XC3.3-64}{(Kernels of matrices)}
Prove or disprove that $\ker(A)=\ker(B)$.
$$
 A=\left(\begin{array}{rrrrr}
    0 &  0 &  0 & 0 & 0  \\
    1 &  0 &  1 & 0 & 0 \\
    3 &  3 &  1 & 3 & 0 \\
    5 &  3 &  1 & 3 & 0
   \end{array}\right), \quad
 B=\left(\begin{array}{rrrrr}
    0 &  0 &  0 & 0 & 0 \\
    1 &  0 &  1 & 0 & 0 \\
    3 &  3 &  1 & 3 & 0 \\
    5 &  3 &  1 & 3 & 0
   \end{array}\right).
$$

\Prob{XC3.4-18}{(Coordinates and spanning sets)}
Let $V=\span\{ \.v_1,\.v_2,\.v_3\}$, where the vectors are displayed below.
Test for $\.x$ in $V$, and if true, then report the coordinates of $\.x$ relative to the
the vectors $\.v_1$, $\.v_2$, $\.v_3$.
$$
\.v_1=\colvector{1 \\ 1\\ 1 },\quad
\.v_2=\colvector{1 \\ 2\\ 3},\quad
\.v_3=\colvector{1 \\ 3\\ 6}, \quad
\.x=\colvector{7 \\ 1\\ 3}.
$$

\Prob{XC3.4-30}{(Matrix of a linear transformation)}
Find the matrix $B$ of the linear transformation $T$ relative to the
basis $\.v_1$, $\.v_2$, $\.v_3$.
$$
T(\.x)= \threebythree{5 & -4 & -2}{-4 & 5 & -2}{-2 & -2 & 8},
$$
$$
\.v_1=\colvector{2 \\ 2\\ 1 },\quad
\.v_2=\colvector{1 \\ -1\\ 0},\quad
\.v_3=\colvector{0 \\ 1\\ -2}.
$$


\Prob{XC3.4-46}{(Basis of a plane)}
Consider the plane $2x_1-3x_2+4x_3=0$. Choose a basis $\.v_1$, $\.v_2$
for the plane, arbitrarily, your choice. Then determine the vector $\.x$
which has coordinates $2$, $-1$ relative to this basis.

{\bf End of extra credit problems chapter 3.}

\end{document}
