ALA K+Kf+Bt, part a, fall 2002
Program
1. Create the   2 x 2   matrix   A  with entries  A(1, 1) = 1,  A(1, 2) = 2, 
A(2, 1) = 3  and  A(2, 2) = 4.
2. Define vectors  e1 = [1; 0]  and  e2 = [0; 1].  Compute  A*e1  and  Ae2. 
Explain why  A*e1  equals the first column of  A,  and  A*e2  the
second.
3. Compute, using your "head computer", possibly supported
by pen and paper, the product  A*v  with  v = [1; -2].  Then check
your answer using matlab.
4. Does the equation  Ax = b  have a (unique) solution (vector)
 x = [x1; x2]  for any given vector  b = [b1; b2] ? Motivate your answer.
Hint: recall a lecture about a_1 x a_2 being nonzero.
5. Find (using pen and paper calculation) the solution of  Ax = b 
with  b = [6; 10].  Then check your answer by computing  A*x  using
matlab.
6. Note that  Ax  in 5 equals  Av  in   3   scaled by the factor   -2. 
Is their a corresponding relation between  x  and  v ?  Is this a
coincidence or not? Hint: recall a lecture about   f(x) = Ax  being linear.
7. With  A  and  b  as in   5,  compute  A \ b  (the famous matlab
"backslash"). What do you get? (hint: compair to   x  in 5)
8. Now define the matrix  C = [1 2; 3 6].  Seek   x   such that
 Cx = d  with   a)  d = [2; 3]  b)  d = [1.5; 4.5].  Conclusions? How does matlab
handle these equations, using  C \ d ?
9. Compute  AC.  Compare with matlab. Is  A*C = C*A ?  Again, check
with matlab?
10. Create the   2 x 3 - matrix  B  with entries  B(i, j) = 2 * i - j  in row
  i  and column   j.  Compute  AB  and try to compute  BA.  Conclusions?
11. Create the matrix  D = [A; C].  Then check   size(D),  size(D, 1) 
and   size(D, 2).  If you are uncertain about the interpretation,
consult help size. What is the size of  DB ?  Check by computing
 D*B!
12. Compute  A', B', D'  and  D''. A'  is called the transpose of
 A.  How is  A'  denoted in AMB&S?
Check if  [1 2] == [1, 2].  Check if  [1 2] == [1; 2].  Check if
 [1 2] == [1; 2]'.
13. Try the special commands   zeros(m, n)  and   ones(m, n)  with   m = 2 
and   n = 3,   say.
Define  F(2, 3) = 7.  What is  F(i, j)  for other combinations of   i
  and   j   than  (2, 3) ?
Also, try   eye(n),  with   n = 2   and   3   say. Define
 I = eye(2).
14. Seek a matrix  X  such that  AX = I.  First try the following
method. Given  A  as before, define
 X = [A(2, 2) - A(1, 2); -A(2, 1) A(1, 1)] / (A(1, 1)*A(2, 2) - A(1, 2)*A(2, 1)). 
Check that  A*X = I  by computing  A*X.  Check if also  X*A = I.  What
is the matrix  X  called, and how is it denoted in this case?
15. Compute  Y = A \ I  and compare to  X.
16. Solve  A*x1 = e1  and  A*x2 = e2  and define  Z = [x1 x2].  How does
 Z  compare to  X  and  Y ?  Do you see why ?
17. Define vectors  a = [1 2 3]  and  b = [4 5 6].  Compute the vector
product  a x b = [ a(2)*b(3) - a(3)*b(2), a(3)*...
18. Compare the result in 17 to writing  cross(a, b).  Compare
  cross(a, b)  to   cross(b, a).
19. Compute the scalar product  a.b  of  a  and  b,  first as
 a(1)*b(1)+.. ,  then as   sum(a.*b),  then as  a*b',  then as   dot(a, b). 
Make sure you understand how each of these alternatives work.
20. Compute the projection of  a  onto  b,  and the projection of
 b  onto  a,  respectively.
21. What is the area of the triangle with vertices in the origin
and in   (2, 1)  and in   (-2, 3) ?  Check your answer is reasonable by
drawing a figure and estimating the area.
22. What is the length   |v|  of the vector  v = [-3 4] ?  Compute both
by pen and paper and using matlab. Recall that   |v| = sqrt(v*v'). 
Also, compare to   norm(v).  What does   length(v)  give in matlab?
Check   help length  (and recall size).
23. What is the angle between the vectors   a)  v  and
 w = [8 6]  b)  v  and  u = [2 1] ?  (hint:  v.w = |v| |w| cos(x)   so that
cos(x) = v.w / (|v| |w|).  In   b)  you may solve this equation using
your Bisection solver, or check out help acos.
24. Try the following matlab function for drawing a vector  a =
(a(1), a(2)).
function   arrow(a)
n = [ -a(2) a(1)];
X = [ 0 a(1) .9*a(1)+.05*n(1) a(1) .9*a(1)-.05*n(1)];
Y = [ 0 a(2) .9*a(2)+.05*n(2) a(2) .9*a(2)-.05*n(2)];
line(X, Y)
set(gca, 'XLim', [-1 1]*max(abs(X)), 'YLim', [-1 1]*max(abs(Y)))
axis('equal')
Save as arrow.m. Then try it out for different vectors a, for
example arrow([2 1]) and then arrow([-1 2]). What do you then
see?
25. Maybee you then want to make a vector move around, for example
by a matlab function of the form
function CounterClock()
figure(1)
set(gcf, 'DoubleBuffer', 'on')
for v = 0 : .1 : 10
a = [cos(v) sin(v)];
arrow(a)
set(gca, 'XLim', [-2 2], 'YLim', [-2 2])
drawnow
cla
end
/Kenneth
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