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  Hour~Exam~~1  --- Math 310 (Applied~Linear~Algebra) ---  1 PM section --- February 18, 2005


{\bf Show all of your work!} An unjustified answer is not correct.\\
Put all of your work and answers on the blank paper handed out.
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{\bf \#1)}  	(20 points)   Given the equations
\begin{eqnarray*} 
 x +2y + 3 z - 3 w & = & 1\\
4 x + 5 y + 6 z  - 6 w & = & 1\\
7 x  + 8 y + 9 z - 8  w& = & 1
\end{eqnarray*}
a)   Give the Reduced Row Eschelon form of the associated augmented matrix.  

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b) Which are the free variables? Which are the dependent variables?
 
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c)  Give the general solution of the system of equations. 
  
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{\bf \#2)} (15 pts)	 Let $\displaystyle A = \left[ \begin{array}{cccc} 2 & 1 & 3 & 0\\4 & 2 & 6 & 2 \\6 & 3 & 9 & 3
\end{array}\right]$
 
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a)  Find a basis   for the {\bf null space} of the matrix $A$


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b) Find a basis   for the {\bf column space} of the matrix $A$
 
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 Show your work - you must justify your answers!
 
 

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 {\bf \#3)}  (20 pts) {\bf Use Cramer's Rule} to solve for the value of $\bf z$   in the solution of the system of equations

$$ \begin{array}{rcrcrcr}  x & + &  y & +  & z  & = & 0\\
 x & + & 2  y & + & 4 z & = & 8\\
 x & + &  3 y & + & 9 z & = & 2
\end{array}$$

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{\bf \#4)} 	(20 pts) Let $C$ be  given by 
$\displaystyle C =  \left[ \begin{array}{rrr}
1 &   2 &   3 \\
2 &   2 &   3 \\
3 &   4 &   4
\end{array}\right]
$.

a) Find the inverse matrix $C^{-1}$. 
You can use either the method of Gaussian Elimination, or the cofactor method - but show all your work!  

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b) Check your answer by multiplying $C$ by your solution for  the inverse  $C^{-1}$.



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{\bf \#5)} (15 pts) 
 Compute the ``LU'' factorization  of the matrix 
$\displaystyle C =  \left[ \begin{array}{rrr}
1 &   2 &   3 \\
2 &   2 &   3 \\
3 &   4 &   4
\end{array}\right]
$.
  
  
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{\bf \#6)} (10 pts) Is the set $\displaystyle \left\{     \left[ \begin{array}{c}
x \\
x^3
\end{array}\right] {\rm  where } ~  x \in {\mathbb R} \right\}$
a vector subspace of   ${\mathbb R}^2$? Justify your answer.

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