\magnification=\magstep2
\nopagenumbers
\centerline{Test III for Calculus II, Math1502, March 14, 2000}
\bigskip
{\bf Name:}
\bigskip{\bf This test is to be taken without graphing calculators, but a
one--page summary of the relevant topics may be used}. 
The allowed time is $50$ minutes. Write answers
in boxes where provided.
Provide exact answers; not decimal approximations! For example, if you mean
$\sqrt 2$ do not write $1.414 \dots$. Please show all your work because
otherwise credit cannot be given.
\vskip 5.9 true in

Topics: 

Given a matrix find a basis of the null space, a basis for the 
column space and their respective dimensions.

Find the nullspace and the column space in terms of planes and lines.

Find the shortest distance of a point and a subspace.

Given a linear transformation on some vectors. They should find the
associated matrix.

General solution = particular solution plus all the solutions of the
homogeneous equation.

Start with a series of simple questions:

A system of equations
$$
A \vec x = \vec b
$$
has a solution
$$
\left [\matrix{ \cr \cr \cr}\right]
$$ 
and the null space is spanned by the vectors ...

Find all the soutions of the system.

Which of the following set of vectors are linearly independent, linearly
dependent. Which forms a basis of $R^2$, which a basis of $R^3$.

The differential equation ... has the particular solution ... and
all the solutions of the homogeneous equation are given by ...
Find the general solution, i.e., all the solution for the differential
equation.

True or false: 

The null space and the column space of a matrix are 
always perpendicular to each other

The number of column vectors is always bigger than the dimension of the
column space.

The dimension of the nullspace of a matrix $A$ and the dimension of the
column space of the adjoint matrix $A^{T}$ always ad up to the number
of column vectors of $A$.

The dimension of the Null Space $N(A)$  is the same as the dimension
of $N(A^{T})$.

A $4 \times 3$ matrix $A$ has a Null Space $N(A)$ whose dimension is $2$. 
Find the dimensions of the Column Space $R(A)$, the dimensions of 
$N(A^{T})$ and the dimension of $R(A^{T})$.

Given the matrix 
$$
A=\left[ \matrix{ 1\  0\  3 \cr 2\  1\  2 \cr 3\  1\  5 \cr} \right] \ .
$$  
Find the the column space of that matrix? Give its equation if it is
a plane, give its parametric representation if it is a line.

Given the matrix 
$$
A=\left[ \matrix{ 1\  {1 \over 2}\  {3 \over 2} \cr 3\  {3 \over 2}
\  {9 \over 2} \cr 2\  1\  3 \cr} \right] \ .
$$  
Find the the column space of that matrix? Give its equation if it is
a plane, give its parametric representation if it is a line.

Find the vector $\vec x$ in ${\cal R}^3$ of smallest length that has its tip on the 
intersection of the planes $x+y+z=1$ and $x-y-z=2$.

Find the vector $\vec x$ in ${\cal R}^3$ of smallest length that has its tip on the 
intersection of the planes $x+y+z=1$ and $x-y-z=2$.


{\bf I: } The null space of a $4 \times 4$ matrix $A$ is spanned by the
vectors
$$
\left [ \matrix{ 2 \cr -1 \cr -4 \cr 3 \cr} \right]
\left [ \matrix{ 3 \cr 4 \cr -1 \cr -2 \cr} \right]
\left [ \matrix{ -4 \cr 3 \cr -2 \cr 1 \cr} \right]
$$
Are these vectors linearly independent?
Find the column space of the transposed matrix of $A$, $A^{T}$.

\bigskip
\noindent
{\bf II: (30 points)} Of the following matrices, find the dimensions of
the column spaces and the dimensions of the null spaces.  



\bigskip
\noindent
{\bf III: (20 points)}


\bigskip
\noindent
{\bf IV: (25 points)} Find the least square solution of the 
overdetermined system $A\vec x =\vec b$ where
$$
A= \left[\matrix{2\ 1 \cr 4\ 2 \cr 1\ 1 \cr}\right] \  ,\ \vec b =\left[\matrix{
1 \cr 0 \cr 0 \cr} \right] \ .
$$


Consider the matrix $A$ and the vector $\vec b$.
$$
A= \left[\matrix{1\ 3 \cr 2\ 2 \cr 3\ 1 \cr}\right] \  ,\ \vec b =\left[\matrix{
1 \cr 0 \cr 0 \cr} \right] \ .
$$
Is $\vec b$ in the column space of the matrix $A$? Compute the least square
solution of the system $A\vec x = \vec b$.

\end
