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\begin{document} 
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\noindent
{\Large Math 211 \hfill Homework 3 \hfill Texfile}

\medskip
Five points for each question.


\medskip
\noindent{\bf 1.}
Determine whether the following three vectors are linearly independent.
{\footnotesize
$$\left[\matrix{0\cr 2\cr 3}\right], \  \left[\matrix{0\cr 0\cr -8}\right], \ 
\left[\matrix{-1\cr 3\cr 1}\right].$$}


\medskip
\noindent{\bf 2.}
Determine the value(s) $h$ such that the following three vectors are linearly independent.
$${\bf v_1} = {\footnotesize\left[\matrix{3\cr -6\cr 1\cr}\right]}, \
{\bf v_2} = {\footnotesize\left[\matrix{-6\cr 4\cr -3\cr}\right]}, \ 
{\bf v_3} = {\footnotesize\left[\matrix{9\cr h\cr 3\cr}\right]}.$$


\medskip
\noindent{\bf 3.} Determine (with explanation) the reduced echelon form of a 
$3 \times 3$ matrix $A$ if it has linearly independent 
columns. 


\medskip
\noindent{\bf 4.}
Let 
{\footnotesize 
$$A = \left[\matrix{1 & 2 & 3\cr 0 & 4 & 5 \cr
0 & 0 & 6}\right],  {\bf u} = \left[\matrix{3\cr 6\cr -9\cr} \right], 
{\bf v} = \left[\matrix{a\cr b\cr c\cr}\right].$$} 
Determine $T(\bu)$ and $T(\bv)$ if 
$T: \IR^3 \rightarrow \IR^3$ is the linear transformation defined by $T(\bx) = A\bx$.



\medskip
\noindent{\bf 5.}
Let
{\footnotesize
$$A = \left[\matrix{1&-2&3\cr 0&1&-3\cr 2&-5&6}\right], \quad {\bf b} = 
\left[\matrix{-6\cr -4\cr -5\cr}\right].$$}
Find a vector ${\bf x}$ such that $A{\bf x}={\bf b}$.



\medskip
\noindent{\bf 6.}
Let  
{\footnotesize
$$A=\left[\matrix{3&2&10&-6\cr 1&0&2&-4\cr 0&1&2&3\cr 1&4&10&8\cr}\right].$$}
Find all vectors that are mapped into the zero vector
by the transformation ${\bf x} \rightarrow A{\bf x}$. 

\medskip
\noindent{\bf 7.}
Let
{\footnotesize
$${\bf x}=\left[\matrix{x_1\cr x_2}\right], 
{\bf v_1}=\left[\matrix{-3\cr 5}\right],
{\bf v_2}=\left[\matrix{7\cr -2}\right],$$}
and let $T: \IR^2 \rightarrow \IR^2$ 
such that $T({\bf x}) = x_1{\bf v_1}+x_2{\bf v_2}$. 
Find the matrix $A$ such that $T({\bf x}) = A{\bf x}$ for each 
${\bf x} \in \IR^2$.



\medskip
\noindent{\bf 8.}
Find the standard matrix of a transformation $T$ that 
reflects points through the horizontal $x_1$ axis and then reflects the 
point through the line $x_2=x_1$. 

[Hint: Determine $T(\be_1), T(\be_2)$.



\medskip
\noindent{\bf 9.}
Suppose $T(x_1, x_2)=(x_1+4x_2, 0,x_1-3x_2,x_1).$ 
Find a matrix $A$ such that
$$A\left[\matrix{x_1\cr x_2\cr}\right] = 
{\footnotesize
\left[\matrix{x_1+4x_2\cr0\cr x_1-3x_2\cr x_1\cr}\right]}.$$ 

[Hint: Study Example 5 in Section 1.9.]
 


\end{document}

