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\begin{document}
\begin{center}
{\bf Homework 6}\\
(due Wednesday, October 5)
\end{center}
\vspace*{.3in}

\noindent{\bf Book Problems:} 
\begin{description}
\item[2.3 (pages 51-52] 3.4, 3.6
\item[2.7 (pages 66-68)] 7.2, 7.3 (don't worry about finding a basis for the nullspace of $A^T$).
\item[2.8 (pages 72-73)] 8.3, 8.4 
\end{description}

\begin{prob}
In Problem 3 on homework 4, consider each of the parts in which the transformation turned out to be linear.  In each case, describe the kernel of the transformation and say whether the transformation is injective.  Also determine whether the transformation is surjective, and if not give an explict counterexample.
\end{prob}

\begin{prob}
Given a set of vectors $S = \{\vv_1,\vv_2,\dots,\vv_6\}\in \R^4$, we form a 
(not augmented) matrix using $\vv_j$ as the $j$th column.  After a bunch of
elementary row operations, we arrive at the following reduced echelon form for
the matrix.
$$
\left[\begin{matrix}
1 & 0 & 1 & 0 & 0 & 2 \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 2 & 0 \\
0 & 0 & 0 & 0 & 0 & 0
\end{matrix}\right]
$$  
State the dimension of $\spn S$ and give a basis?  
Explain your answer.
\end{prob}

\begin{prob}
Let $A\in M_{n\times n}(\R)$ be a square matrix.  In order to strike fear in the hearts of future students of linear algebra, use `existing technology' (i.e. stuff we've already discussed in class), write down as many statements as you can that are logically equivalent to `$A$ is invertible.' (5 statements gives small children momentary pause, 10 causes significant fraction of novices to stop reading and watch latest American Idol rerun, 15 triggers ND alert system and makes local news, 20+ causes global panic followed by revolt and then complete suppression of all mathematics everywhere.)
You need not justify your answer.  Your time will be spent more profitably seeking immediate cover.
\end{prob}

\begin{prob}
This problem concerns two linear transformations $T:\R^3\to\R^3$.  In each case, do the following:
\begin{itemize}
\item Find the `standard' matrix $A:=[T]_{\cE\cE}$ by computing the columns $T(\ve_j)$ of $A$ directly using formulas for orthogonal projections, etc, etc (I'll forgive you for not doing this part if you aren't in Connolly's class and haven't worked with orthogonal projections before)
\item Find $A$ (again) by first computing the matrix for $T$ relative to a `nice' basis and then using change of coordinates.
\item Use $A$ to compute $T(1,2,3)$.
\end{itemize}
Note that you should use the dot product as your inner product on $\R^3$.
\begin{enumerate}
\item $T$ projects vectors orthogonally onto the plane $x-2y+z = 0$.
\item $T$ reflects vectors through the line $\spn\{(1,1,1)\}$.
\end{enumerate}
\end{prob}

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