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\begin{flushright}
  Your Name Goes Here
\end{flushright}


\begin{center}
  \textbf{Math 24}\\ Homework 1
\end{center}



\begin{itemize}


\item [\#1.2.4a] Find the RREF of the matrix
  $\left[\begin{array}{rrr}
3 & 0 & 2 \\
1 & -4 & 1
\end{array}\right]$.

\item [\#1.2.4c]Find the RREF of the matrix
  $\left[\begin{array}{rrr}
-3&0\\1&2\\4&-1
\end{array}\right]$.


\item [\#1.2.6d] Find all the solutions of the linear system
  \begin{align*}
    3x+y-2z&=-3\\
    x+0y+2z&=4\\
    -x+2y+3z&=1\\
    2x-y+z&=-6
  \end{align*}


\item [\#1.2.6e] Find all the solutions of the linear system
  \begin{align*}
    3x+y-2z&=-3\\
    x+0y+2z&=-4\\
    -x+2y+3z&=1\\
    2x-y+z&=-6
  \end{align*}



\item [\#1.2.13] Give examples of linear system of each of the
  following types if possible.  Explain how you know they have the
  properties, or else explain why there is no such system.
    \begin{enumerate} [label=(\alph*)]
    \item Underdetermined and inconsistent.
    \item Underdetrermined with a unique solution.
    \item Underdetermined with more than one solution.
    \item Overdetermined and inconsistent.
    \item Overdetrermined with a unique solution.
    \item Overdetermined with more than one solution.
    \item Square and inconsistent.
    \item Square with a unique solution.
    \item Square with more than one solution.
    \end{enumerate}

  
  \item [\#1.3.10] Consider a linear system in vector form
    \[x_1\v_1 + \cdots + x_n\v_n = \b,
    \]
    where $\v_1, \dots, \v_n, \b \in \R^m$.  Show that the system is
    consistent if and only if\newline $\b \in \la \v_1, \dots,
    \v_n\ra$.

  \item [\#1.3.12] Let $\v_1, \v_2, \v_3 \in \R^3$, and suppose that
    the linear system $x\v_1 + y\v_2 + z\v_3 = \0$ has infinitely many
    solutions.  Show that  $\v_1, \v_2, \v_3$ lie in a plane in $\R^3$
    which contains the origin $\0$.
    
 \item [\#1.4.10] An $n\times n$ linear system over a field $\mathbb
   F$ is called \textbf{upper triangular} if the coefficient $a_{ij} =
   0$ whenever $i>j$:
   \begin{align*}
     a_{11}x_1 + a_{12}x_2 + a_{13}x_3 + \cdots + a_{1n}x_n &=b_1\\
     a_{22}x_2 + a_{23}x_3 + \cdots+a_{2n}x_n &= b_2\\
     a_{33}x_3 + \cdots + a_{3n}x_n &= b_3\\
     \ddots\hspace{.8in}          &\vdots\\
     a_{nn}x_n &= b_n.
   \end{align*}
Show that if $a_{ii}\ne 0$ for each $i$, then the system is consistent
with a unique solution.

 \item [\#1.5.4] Determine which of the following subsets are
   subspaces of $C[a,b]$.  Here $a, b$ are fixed real numbers and $a <
   c< b$.
   \begin{enumerate}[label=(\alph*)]
   \item $V = \{ f\in C[a,b]\mid f(c) = 0\}$.
   \item $V = \{ f\in C[a,b]\mid f(c)=1\}$.
   \item $V = \{ f\in D[a,b]\mid f'(c)=0\}$.
   \item $V = \{ f\in D[a,b]\mid f' \mbox{ is constant}\}$.
   \item $V = \{ f\in D[a,b]\mid f'(c)=1\}$.
   \end{enumerate}

 \item [\#1.5.11] Show that if $U_1$ and $U_2$ are subspaces of a
   vector space $V$, then
   \[U_1+U_2 := \{u_1+u_2\mid u_1\in U_1, u_2\in U_2\}
     \]
is also a subspace of $V$.
    
\end{itemize}

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