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\headers{Math343}{Homework 3}{Due Sep 10, 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 {}[1.3.\{11,12\}] Determine if $\vect{b}$ is a linear combination of $\vect{a_1}$, $\vect{a_2}$, and $\vect{a_3}$.
\begin{enumerate}
	\item $\vect{a_1} = \MTBO{1}{-2}{0}$, $\vect{a_2} = \MTBO{0}{1}{2}$, $\vect{a_3} = \MTBO{5}{-6}{8}$, $\vect{b} = \MTBO{2}{-1}{6}$.
	\item $\vect{a_1} = \MTBO{1}{-2}{2}$, $\vect{a_2} = \MTBO{0}{5}{5}$, $\vect{a_3} = \MTBO{2}{0}{8}$, $\vect{b} = \MTBO{-5}{11}{-7}$.
\end{enumerate}

\item {}[1.3.13] Determine if $\vect{b}$ is a linear combination of the vectors formed from the columns of the matrix $A$.
\begin{align*}
A &= %
\left[%
\begin{array}{*{3}{r}}
1 & -4 & 2\\
0 & 3 & 5\\
-2& 8 & -4
\end{array}
\right]%
%
&
%
\vect{b} &= \MTBO{3}{-7}{-3}
\end{align*}

\item {}[1.3] True/False.  Justify your answers.
\begin{enumerate}
	\item Another notation for the vector $\MTWBO{-4}{3}$ is $[-4~3]$.
	\item The points in the plane corresponding to $\MTWBO{-2}{5}$ and $\MTWBO{-5}{2}$ lie on a line through the origin.
	\item An example of a linear combination of vectors $\vect{v_1}$ and $\vect{v_2}$ is the vector $\frac{1}{2}\vect{v_1}$.
	\item The weights $c_1,\ldots,c_p$ in a linear combination $c_1\vect{v_1} + \cdots + c_p\vect{v_p}$ cannot all be zero.
	\item The set $\Span\{\vect{u},\vect{v}\}$ is always visualized as a plane through the origin.
\end{enumerate}

\item {} Pivot columns in a matrix.
\begin{enumerate}
	\item Let $A = [~\vect{a_1}~\vect{a_2}~\cdots~\vect{a_p}~]$ and let $B = [~\vect{b_1}~\vect{b_2}~\cdots~\vect {b_p}~]$.  Suppose that $\vect{a_i} = \vect{a_j}$ and $A$ is row equivalent to $B$.  Explain why it must be that $\vect{b_i} = \vect{b_j}$.  (Hint: how do the $i$th and $j$th columns behave with respect to each elementary row operation?)
	\item Let $A = [~\vect{a_1}~\vect{a_2}~\cdots~\vect{a_p}~]$.  Suppose that $\vect{a_i} = \vect{a_j}$ and that $i<j$.  Use part (a) to explain why the $j$th column of $A$ cannot be a pivot column.  (Hint: let $B$ be the reduced echelon form of $A$.)
\end{enumerate}

\item {} [1.4.9] Write the system first as a vector equation and then as a matrix equation.
\[
\begin{array}{*{7}{r}}
	3x_1  &+&  x_2  &-& 5x_3  &=& 9\\
	      &&   x_2  &+& 4x_3  &=& 0
\end{array}
\]

\item {} [1.4.11] Given $A$ and $\vect{b}$ below, write augmented matrix corresponding to the matrix equation $A\vect{x} = \vect{b}$ and solve for $\vect{x}$.
\begin{align*}
A &= \left[%
\begin{array}{*{3}{r}}
1 & 2 & 4\\
0 & 1 & 5\\
-2 & -4 & -3
\end{array}
\right]%
%
&
%
B &= \MTBO{-2}{2}{9}
\end{align*}

\end{enumerate}
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