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


\begin{center}
\bf Math 2270 Extra Credit Problems \\
\bf Chapter 5 \\
{\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{Xc5.1-10}{(Angle)}
For which values of $k$ are the vectors $\.u=\colvector{2k \\ 1 \\ 3}$ and $\.v=\colvector{1\\ k \\ 2}$ perpendicular?

\Prob{Xc5.1-26}{(Orthogonal Projection)} Find the orthogonal projection of  $\.w$ onto the subspace $V$, given
$$\.w=\colvector{1\\ 0\\ 0\\ 0},\quad V=\span\left\{
\colvector{1\\ 1\\ 1\\ 1},\colvector{1\\ 1\\ -1 \\ -1}, \colvector{1\\ -1 \\ -1\\ 1}\right\}.$$

\Prob{Xc5.1-34}{(Minimization)} Among all the vectors $\colvector{x\\ y\\ z}$ in $\Rr^3$, find the one with unit length that minimizes the sum $x+2y+3z$.

\Prob{Xc5.2-14}{(Gram-Schmidt Basis)} Given the basis below, labeled $\.v_1$, $\.v_2$, $\.v_3$, find the Gram-Schmidt basis $\.u_1$, $\.u_2$, $\.u_3$.
$$\colvector{1\\ 7\\ 1 \\ 7}, \colvector{0\\ 7\\ 2\\ 7}, \colvector{2\\ 15\\ 2\\ 13}.$$

\Prob{Xc5.2-20}{(QR-Factorization)} Find the factorization $M=QR$, given
$$M=\threebythree{4&25&0}{0&0&-2}{3&-25&0}.$$

\Prob{Xc5.2-34}{(Kernel)} Find an orthonormal basis for the kernel of the matrix $A=\left(\begin{array}{llll} 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 \end{array}\right)$.

\Prob{Xc5.2-38}{(QR-Factorization)} Find the factorization $M=QR$, given
$M=\left(\begin{array}{rrr}  0&-3&0 \\ 0&0&0 \\ 4&0&0 \\ 0&0&5 \end{array}\right)$

\Prob{Xc5.3-11}{(Orthogonal Matrices)}  Given $A$ and $B$ are orthogonal, then which of the following must be orthogonal?

\noindent (a) $2A$, (b) $ABA$, (c) $A^{-1}B^T$, (d) $A-AB$, (e) $AB+BA$, (f) $-BA$

\Prob{Xc5.3-20}{(Symmetric Matrices)} Given $A$ and $B$ are symmetric matrices and $A$ is invertible, then which of the following must also be symmetric?

\noindent (a) $A^TA$, (b) $ABA$, (c) $A^{-1}B$, (d) $A-B$, (e) $A-BA$, (f) $A-A^T$, (g) $A^TB^TBA$, (h) $B(A+A^T)B^T$

\Prob{Xc5.3-26}{(Dot Product)} Let $T$ be an orthogonal transformation from $\Rr^n$ to $\Rr^n$.
Prove that $\.u\cdot\.v=T(\.u)\cdot T(\.v)$ for all vectors $\.u$ and $\.v$ in $\Rr^n$.

\Prob{Xc5.3-32a}{(Orthogonal Matrices)} Assume $A$ is $n\times m$ and $A^TA=I$. Is $AA^T$ the identity matrix? Explain.


\Prob{Xc5.3-44}{(Orthogonal Matrices)} Consider an $n\times m$ matrix $A$. Find in terms of $n$ and $m$ the value of the sum $\dim(\image(A))+\dim(\ker(A^T))$.

\Prob{Xc5.3-50}{(QR-Factorization)} ~\newline

\noindent (a) Find all square matrices $A$ that are both orthogonal and upper triangular with positive diagonal entries.

\noindent (b) Show that the $QR$-factorization is unique for an invertible square matrix $A$. Hint: see Exercise 50b in {\bf Bretscher 3E}, section 5.3.

\Prob{Xc5.4-5}{(Basis of $V^\perp$)} Find a basis for $V^\perp$, where $V=\ker(A)$ and
$$A=\left(\begin{array}{llll} 1&1&1&1 \\ 1&2&4&3 \\ 0&0&0&0 \end{array}\right).$$

\Prob{Xc5.4-16}{(Rank)} Prove or disprove: The equation $\rank(A)=\rank(A^TA)$ hold for all square matrices $A$.


\Prob{Xc5.4-22}{(Least Squares)} Find the least squares solution $\.x^*$ of the system $A\.x=\.b$, given
$$A=\left(\begin{array}{ll} 3&2\\ 5&3\\ 4&5 \end{array}\right),\quad \.b=\colvector{10\\ 18\\ 4}.$$

\Prob{Xc5.4-26}{(Least Squares)} Find the least squares solution $\.x^*$ of the system $A\.x=\.b$, given
$$A=\left(\begin{array}{lll} 1&2&3\\ 4&5&6\\ 7&8&9 \end{array}\right),\quad \.b=\colvector{2\\ 0\\ 0}.$$

\Prob{Xc5.5-10}{(Orthonormal Basis)} Find an orthonormal basis for $V^\perp$, where $V=\span\{ 1+t^2\}$, in the space $W$ of all polynomials $a_0+a_1t+a_2t^2$ with inner product
$<f,g>=\frac12\int_{-1}^1f(t)g(t)dt$.

\Prob{Xc5.5-24}{(Orthonormal Basis)} Consider the linear space $P$ of all polynomials with inner product $<f,g>=\int_{0}^1f(t)g(t)dt$. Let $f$, $g$, $h$ be three polynomials satisfying the relations
$$
\begin{array}{lll}
<f,f>=4 & <f,g>=0 & <f,h>=8 \\
<g,f>=0 & <g,g>=2 & <g,h>=4 \\
<h,f>=8 & <h,g>=4 & <h,h>=10
\end{array}
$$

\noindent (a) Find $<f,g+2h>$.

\noindent (b) Find $\| g+h\|$.

\noindent (c) Find $c_1$, $c_2$ satisfying $\proj_{\span\{f,g\}}(h)=c_1f+c_2g$.

\noindent (d) Find an orthonormal basis for the span of $f$, $g$, $h$ expressed as
linear combinations of $f$, $g$ and $h$.

\vfill

{\bf End of extra credit problems chapter 5.}

\end{document}
