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\headers{Math343}{Homework 9}{Due Oct 22, 2025}

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\noindent
\textbf{Directions:} You may work to solve these problems in groups, but all written work must be your own.  Show all work; no credit for solutions without work..

\begin{enumerate}


\item {}[2.8.\{15-17\}] Determine which sets below are bases for $\RR^2$ or $\RR^3$.  Justify each answer.
\begin{enumerate}
	\item $\kgmmatrix{1}{5 \\ -2}, \kgmmatrix{1}{10 \\ -3}$ 
	\item $\kgmmatrix{1}{-4 \\ 6}, \kgmmatrix{1}{2 \\ -3}$ 
	\item $\kgmmatrix{1}{0 \\ 1 \\ -2}, \kgmmatrix{1}{5 \\ -7 \\ 4}, \kgmmatrix{1}{6 \\ 3 \\ 5}$
\end{enumerate}

\item {}[2.8.\{23-25\}] Given a matrix $A$ and an echelon form of $A$, find a basis for $\Col(A)$ and $\Nul(A)$.
\begin{enumerate}
	\item $A=\kgmmatrix{4}
	{
		4 & 5 & 9 & -2 \\
		6 & 5 & 1 & 12 \\
		3 & 4 & 8 & -3
	}
	\sim
	\kgmmatrix{4}
	{
		1 & 2 & 6 & -5\\
		0 & 1 & 5 & -6\\
		0 & 0 & 0 & 0
	}$
	
	\item $A=\kgmmatrix{4}
	{
		-3 & 9 & -2 & -7 \\
		2 & -6 & 4 & 8 \\
		3 & -9 & -2 & 2
	}
	\sim
	\kgmmatrix{4}
	{
		1 & -3 & 6 & 9 \\
		0 & 0  & 4 & 5 \\
		0 & 0 & 0 & 0
	}$
	
	\item $A=\kgmmatrix{5}
	{
		1 & 4 & 8 & -3 & -7\\
		-1 & 2 & 7 & 3 & 4\\
		-2 & 2 & 9 & 5 & 5\\
		3 & 6 & 9 & -5 & -2
	}
	\sim
	\kgmmatrix{5}
	{
		1 & 4 & 8 & 0 & 5\\
		0 & 2 & 5 & 0 & -1\\
		0 & 0 & 0 & 1 & 4\\
		0 & 0 & 0 & 0 & 0
	}$
\end{enumerate}
 
\item {}[2.8.\{21,22\}] True/False.  Justify your answers.
\begin{enumerate}
	\item A subspace of $\RR^n$ is any set $H$ such that (i) the zero vector is in $H$, (ii) $\mb u, \mb v$, and $\mb u + \mb v$ are in $H$, and (iii) $c$ is a scalar and $c\mb u$ is in $H$.
	\item If $\mb v_1,\ldots, \mb v_p$ are in $\RR^n$, then $\Span\{\mb v_1,\ldots, \mb v_p\}$ is the same as the column space of the matrix $[\mb v_1 \cdots \mb v_p]$.
	\item The set of all solutions of a system of $m$ homogeneous equations in $n$ unknowns is a subspace of $\RR^m$.
	\item The columns of an invertible $n\times n$ matrix form a basis for $\RR^n$.
	\item Row operations do not affect linear dependence relations among the columns of a matrix.
	\item A subset $H$ of $\RR^n$ is a subspace if the zero vector is in $H$.
	\item Given vectors $\mb v_1,\ldots,\mb v_p$ in $\RR^n$, the set of all linear combinations of these vectors is a subspace in $\RR^n$.
	\item The null space of an $m\times n$ matrix is a subspace of $\RR^n$.
	\item The column space of a matrix $A$ is the set of solutions of $A\mb x = \mb b$.
	\item If $B$ is an echelon form of a matrix $A$, then the pivot columns of $B$ form a basis for $A$.
\end{enumerate}

\end{enumerate}
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