CE_Feb11.tex

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\newcommand{\rank}{\textrm{rank}\,}
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\begin{document}

\begin{center}
	\Large
	\rm{Math 221}
	\\
	\rm{Class Exercises:  Feb. 9}
	\\
\end{center}
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\begin{enumerate}


\item {Consider the following linear system
	\begin{eqnarray*}
		x_1 - 4x_2 &=& 1 \\
		3x_1 + hx_2 & = & k
	\end{eqnarray*}
	
	\begin{enumerate}
		\item{Give all values of $h$ and $k$ such that there is no solution to the system.}
		\item{Give all values of $h$ and $k$ such that there is a unique solution to the system.}
	\end{enumerate}
}

\item {Is the set of polynomials $\{x^3+1, x^3-x, x^2 + 1, x+1  \}$ a basis for $\mathbb{P}_3$? Explain why or why not.}

\item {Let $U$ and $W$ be subspaces of a vector space V 
	\begin{enumerate}
		\item{Show that $U\cap W$ is also a subspace.  
			\\(\textit{$U\cap W$ is the set of vectors that are in both $U$ and $W$.})}
		\item{Show by giving a counterexample that in general $U\cup W$ is not a subspace.  \\(\textit{$U\cup W$ is the set of vectors that are in $U$ or $W$.})  } 
	\end{enumerate}
}

\item{Is it possible to determine if the matrix with the following structure must have zero determinant?
\begin{displaymath}
A = \left[ \begin{array}{rrrrr}  
* & * & * & * & * \\  
\** & * & * & * & * \\ 
0 & 0 & 0 & * & * \\ 
0 & 0 & 0 & * & * \\ 
0 & 0 & 0 & * & * \\ 
\end{array} \right] \quad
\end{displaymath}
(The $*$ entries are not equal, just nonzero.)
}

\item{If $B$ and $C$ are invertible matrices, is it true that $\adj{(BC)} = (\adj{B})(\adj{C})$?}

\item{For what vectors $b$ does there exist a solution to $Ax=b$?
\begin{displaymath}
A = \left[ \begin{array}{rrrr} 1 &  2 &  0 & 1 \\ 0 & 0 & 0 & 0\\ 2 & 4 & 0 & 1 \end{array} \right] \quad \quad b = \left [ \begin{array}{r} b_1 \\ b_2 \\ b_3 \end{array} \right ]
\end{displaymath}
\begin{enumerate}
\item{Find a basis for the null space $\mathcal{N}(A)$.}
\item{Find a basis for the column space $\mathcal{C}(A)$.}
\item{In the case that $b$ is in $\mathcal{C}(A)$ write the general form of the solution to $Ax=b$.}
\item{Find the rank of $A^T$}.
\end{enumerate}
}

\item{If $x$,$y \in \mathbb{R}^3$ and $B = xy^T$, what is the rank of $B$? }
\item{If $A \in \mathbb{M}_{3\times 5}$ and $e_1$, $e_2$ and $e_3$ (the standard basis vectors for $\mathbb{R}^3$) are in $\mathcal{C}(A)$, does $A$ have a right inverse?  Does it have a left inverse?}
\item{How many $4\times 4$ permutation matrices are there?  Does this set span $\mathbb{M}_{4\times 4}$?  Is the set linearly independent?}
\item{Show that an $n\times n$ matrix $A$ is invertible if and only if $A^T$ is invertible.  (Can you supply an argument that does not rely on determinants?)}


\end{enumerate}

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