
% extra credit ch4 2250, S2015
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 {\bf Name~}\underline{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}\hfill
 {Class Time~}\underline{~~~~~~~~}\newline


\begin{center}
\bf Math 2250 Extra Credit Problems \\
\bf Chapter 4 \\
{\small\bf S2015}
\end{center}

{\bf Submitted work}. Please submit one stapled package with this sheet
on top. Kindly check-mark the problems submitted and
label the paper \fbox{\bf Extra Credit}. Label each solved problem with
its corresponding problem number, e.g., \fbox{Xc10.3-20}.

\Prob{Xc4.1-20}{(Independence)}
mapleTest independence or dependence:
$\colvector{5\\ 5\\ 4}$,
$\colvector{2\\ 3\\ 1}$,
$\colvector{1\\ -1\\ 2}$.

\Prob{Xc4.1-30}{(Kernel theorem)} Verify from the kernel theorem
(Theorem 2, 4.2) that the set of all vectors in $\Rr^3$ such that
$2x-y=3z$ is a subspace of $\Rr^3$.

\Prob{Xc4.1-32}{(Subspace criterion)} Apply the subspace criterion
(Theorem 1, 4.2) to verify that the set of all vectors in $\Rr^3$
such that $2x-y=3z$, $xy=0$ is \BF{not} a subspace of $\Rr^3$.

\Prob{Xc4.2-4}{(Subspace criterion)}
Apply the subspace criterion (Theorem 1, 4.2) to verify that the set of all
vectors in $\Rr^3$ satisfying $|x|=y+z$ fails to be a subspace of $\Rr^3$.

\clearpage

\Prob{Xc4.2-28}{(Kernel theorem)} Let $S$ be the subset of $\Rr^4$
defined by the equations
$$\twobytwo{1&2}{0&3}\colvector{x_1\\ x_2}=\colvector{0\\ 0}, \quad
\twobytwo{4&0}{1&2}\colvector{x_3\\ x_4}=\colvector{0\\ 0}.
$$
Use the kernel theorem (Theorem 2, 4.2) to prove that $S$ is a subspace
of $\Rr^4$.

\Prob{Xc4.3-18}{(Dependence and frame sequences)}
Give the vectors below, display a frame sequence from the augmented matrix $C$
of the vectors to final frame $\rref(C)$. Use the sequence to decide if the vectors
are independent or dependent. If dependent, then report all possible dependency
relations $c_1\.v_1+c_2\.v_2+c_3\.v_3=\.0$.
$$
\.v_1=\colvector{3\\ 9\\ 0\\ 5},\quad
\.v_2=\colvector{-2\\ 4\\ -10\\ 10},\quad
\.v_3=\colvector{4\\ 7\\ 5\\ 0}.
$$

\Prob{Xc4.3-24}{(Independence in abstract vector spaces)}
Let $V$ be an abstract vector space whose packages of data items $\.v$
have unknown details. You are expected to use only definitions and the
toolkit of 8 properties in the details of this problem.

Assume given $\.v_1$, $\.v_2$, $\.v_3$ independent vectors in $V$. Define
$\.u_1=\.v_1+2\.v_2$, $\.u_2=\.v_3+\.u_1$, $\.u_3=\.v_1+\.u_2$. Prove
that $\.u_1$, $\.u_2$, $\.u_3$ are independent.


\Prob{Xc4.4-6}{(Basis)}
Find a basis for $\Rr^3$ which includes two independent vectors $\.v_1$,
$\.v_2$ which are in the plane $2x-3y+5z=0$ and a vector $\.v_3$ which
is perpendicular to both $\.v_1$ and $\.v_2$. Hint: review the cross
product, a topic from calculus.

\Prob{Xc4.4-24}{(Basis for $A\.x=\.0$)}
Display a frame sequence starting with $A$ having final frame
$\rref(A)$. Use this sequence to find the scalar general solution of
$A\.x=\.0$ and then the vector general solution of $A\.x=\.0$. Finally,
report a basis for the solution space of $A\.x=\.0$.

$$
A=\left(\begin{array}{rrrr}
1 & -4 & -3 &-7 \\
2 & -1 & 1 & 7 \\
1 & 2 & 3 & 11 \\
2 & 6 & 4 & 8
\end{array}\right),\quad
\.x=\colvector{x_1\\ x_2\\ x_3\\ x_4}.
$$

\Prob{Xc4.5-6}{(Row and columns spaces)} Find a basis for the row
space and the column space of $A$, but the bases reported must be
rows of $A$ and columns of $A$, respectively.
$$
A=\left(\begin{array}{rrrr}
1 & -4 & -3 &-7 \\
2 & -1 & 1 & 7 \\
2 & 6 & 4 & 8
\end{array}\right)
$$

\Prob{Xc4.5-24}{(Redundant columns)} Use the pivot theorem
(Algorithm 2, 4.5) to find the non-pivot columns of $A$ (called
redundant columns).
$$
A=\left(\begin{array}{rrrrrr}
1 & -4 & -3 &-7 & 4 & 3\\
2 & -1 & 1 & 7 & 2 & 0\\
1 & 2 & 3 & 11 & 0 & 0\\
2 & 6 & 4 & 8 & 0 & 1
\end{array}\right)
$$

\Prob{Xc4.5-28}{(Rank and the three properties)}
Suppose $A$ is a $5\times 4$ matrix and $A\.x=\.0$ has a basis of size $3$.
What are the possible forms of $\rref(A)$?

\clearpage

\Prob{Xc4.6-2}{(Orthogonality)}
Let $A$ be a $3\times 3$ matrix such that $AA^T=I$. Prove that the columns
$\.v_1$, $\.v_2$, $\.v_3$ of $A$ satisfy the orthogonality relations
$$ |\.v_1|=|\.v_2|=\.v_3|=1, \quad
\.v_1\cdot\.v_2=
\.v_2\cdot\.v_3=
\.v_3\cdot\.v_1=0.
$$

\Prob{Xc4.7-10}{(Subspaces of function spaces)} Let $V$ be the
function space of all polynomials of degree less than 5. Define $S$
to be the subset of $V$ consisting of all polynomials $p(x)$ in $V$
such that $p(0)=p(1)$ and $\int_{-1}^{1} p(x)dx=p(2)$. Prove that
$S$ is a subspace of $V$.

\Prob{Xc4.7-20}{(Partial fractions and independence)} Assume the
polynomials $1$, $x$, \ldots, $x^n$ are \EM{independent}. They are a
basis for a vector space $V$. Use this fact explicitly in the
details for determining the constants $A$, $B$, $C$, $D$ in the
partial fraction expansion
$$
\frac{x^2-x+1}{(x-1)(x+1)^2(x-3)}=\frac{A}{x-1}+\frac{B}{x+1}+\frac{C}{(x+1)^2}+\frac{D}{x-3}.
$$

\Prob{Xc4.7-26}{(Solution space basis for a DE)} Find the general
solution of $3y''+ 5y'=0$, containing symbols $c_1$, $c_2$ for the
arbitrary constants. Take partial derivatives $\partial y/\partial
c_1$, $\partial y/\partial c_2$ to identify two functions of $x$.
Prove that these functions are independent and hence find a basis
for the solution space of the differential equation. Suggestion:
View $3y''+5y'=0$ as two equations $3v'+5v=0$ and $y'=v$. Then use
elementary first order differential equation methods.

\vfill

 {\bf End of extra credit problems chapter 4.}

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