\documentclass{amsart}
\usepackage{amssymb,latexsym} 

\date{}
\begin{document}
\begin{flushright} {\today} \end{flushright}
\begin{center}
{\bf MATH 221 Linear Algebra Quiz V}
\end{center}

\bigskip
\noindent
(I) Decide which of the following statements, concerning an
$m \times n$ matrix $A$, are true and which are false (no
proof is needed however make sure that you understand your
answers).

\medskip
(1) If $B$ is a matrix in echelon form obtained from $A$ via
elementary row operations then the row space of $A$ and the
row space of $B$ are the same.

\medskip
(2) If $B$ is a matrix in echelon form then the non-zero
rows form a basis for the row space of $B$.

\medskip
(3) If $B$ is a matrix in echelon form then the
dimension of the row space of $B$ is equal to the
number of pivots of $B$.

\medskip
(4) If $B$ is a matrix in echelon form then the
dimension of the null space of $B$ is equal to the
number of free variables in the equation $B{\bf x} = 0$.

\medskip
(5) If $B$ is a matrix in echelon form then the
dimension of the row space of $B$ is equal to the
number of leading variables in the equation $B{\bf x} = 0$.

\medskip
(6) If $B$ is a matrix in echelon form then the
columns of $B$ containing the pivots form a basis of the
column space of $B$.

\medskip
(7) If $B$ is a matrix in echelon form obtained from $A$ via
elementary row operations then whenever certain columns of
$B$ form a basis of the
column space of $B$, the corresponding columns of $A$ is a basis of the
column space of $A$.

\medskip
(8) The dimension of the row space of $A$ is the same as the
dimension of the column space of $A$.

\medskip
(9) The sum of the dimension of the row space of $A$ and the
dimension of the null space of $A$ is $n$.

\medskip
(10) The sum of the dimension of the row space of $A$ and the
dimension of the null space of $A$ is $m$.

\medskip
(11) The sum of the dimension of the column space of $A$ and
the dimension of the null space of $A$ is $n$.

\medskip
(12) The sum of the dimension of the column space of $A$ and
the dimension of the null space of $A$ is $m$.

\medskip
(13) The sum of the dimension of the row space of $A$ and the
dimension of the column space of $A$ is $n$.

\medskip
(14) The sum of the dimension of the row space of $A$ and the
dimension of the column space of $A$ is $m$.

\medskip
(15) The rank of $A$ is the dimension of the row space of
$A$.

\medskip
(16) The rank of $A$ is the dimension of the column space of
$A$.

\medskip
(17) The sum of the rank of $A$ and the dimension of the row
space is $n$.

\medskip
(18) The sum of the rank of $A$ and the dimension of the row
space is $m$.

\medskip
(19) The sum of the rank of $A$ and the dimension of the
null space is $n$.

\medskip
(20) The sum of the rank of $A$ and the dimension of the
null space is $m$.

\bigskip
\noindent
(II) Let $A$ be an $n \times n$ matrix which of the following
statements are NOT equivalent to the staement that $A$ is
invertible?

\medskip
(1) $\dim$ Null Space$(A) = 0$

\medskip
(2) $\dim$ Null Space$(A) = n$

\medskip
(3) rank$(A) = n$

\medskip
(4) rank$(A) = 0$

\medskip
(5) $\dim$ Column Space$(A) = 0$

\medskip
(6) $\dim$ Columnl Space$(A) = n$

\medskip
(7) $\dim$ row Space$(A) = 0$

\medskip
(8) $\dim$ row Space$(A) = n$

\bigskip
\noindent
(III) Find the dimension and basis for the null space,
column space and row space of $A$ given below. What is the
rank of $A$?
$$A = \begin{pmatrix}
3&2&-4&1&5\\
6&4&-7&3&1\\
-3&-2&6&1&2\\
9&6&-11&4&6
\end{pmatrix}$$




\end{document}