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\begin{document}

Your name:
\vskip 12pt

Instructor (please circle): 
\hspace{1in}
Alex Barnett
\hskip .7in
Naomi Tanabe
\vskip 12pt

{\bf \large Math 22 Fall 2016, Homework 4, due Wed Oct 12}

{\em Please show your work. No credit is given for solutions without work or justification.}

\vskip .05in

\ben

% 7 pts
\item
\ben
% 4 pts
\item 
Compute the determinant of
$\left[\begin{array}{cccc}1&1&1&1\\3&3&3&1\\6&6&3&1\\10&6&3&1 \end{array}\right]$
\vspace{4in}

% 1 pt for the logic
\item Is the matrix from (a) invertible, and why?

\vspace{0.7in}

%2 pts
\item Let $A$ and $B$ be arbitrary $n\times n$ matrices. Prove that
$\det(BA) = \det(AB)$. (You may use results we have done, but state which.)

\vspace{2in}

\een

\newpage %%%%%%%%%%%%%%%%

% 7 pts
\item 
Is each of the following sets a vector space? Explain what result(s) you
used to prove your claim. You may assume that $\RR^n$ is a vector space.
\ben
\item $\left\{ \left[\begin{array}{c}t\\3\end{array}\right], \quad t
\mbox{ is real } \right\}$

\vspace{1.8in}

\item $\left\{ \left[\begin{array}{c}0\\3+t\end{array}\right], \quad t
\mbox{ is real } \right\}$

\vspace{1.8in}

\item $\left\{ \left[\begin{array}{c}a+b\\-a\\2a-b\end{array}\right],
\quad a \mbox{ and } b \mbox{ are real } \right\}$

\vspace{1.8in}

\item All the vectors in $\RR^4$ for which the first two entries are equal
and the sum of all four entries is zero.

\een

\newpage %%%%%%%%%%%%%%%%

% 6 pts
\item
True or false (no working needed, just circle the answer):
% will be one pt each

\vg

\ben
\itemsep 10ex

\item T / F: \qquad
\parbox{4in}{
$\det(A+B) = \det A + \det B$
}
\item T / F: \qquad
\parbox{4in}{
Given a square matrix,
if you add three times row 2 to row 1, the determinant is multiplied by three.
}
\item T / F: \qquad
\parbox{4in}{
If you double every entry of a $3\times 3$ matrix, its determinant
is multiplied by 8.
}
\item T / F: \qquad
\parbox{4in}{
For any $m\times n$ matrix $A$, if Col $A = \RR^m$ this implies $m\ge n$.
}
\item T / F: \qquad
\parbox{4in}{
For any $m\times n$ matrix $A$, and right hand side $\mbf{b}$ in $\RR^m$,
the solution set of the linear system $A\mbf{x} = \mbf{b}$ is a subspace
of $\RR^n$.
}
\item T / F: \qquad
\parbox{4in}{
The set of all functions on $[0,1]$ with $f(\frac12) = 0$ is a
subspace of the vector space of all functions on $[0,1]$.
}
\een


% boring: $\RR^6$ is a subspace of $\RR^7$.

\een

\end{document}
