Math 464/514-Hotline
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Please look here for answers to questions on homework and other 
class materials, exams etc.
I will post questions (without any identifying info!) and answers
in the reverse order they are received.
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Mon Oct 19 23:07:52 MDT 2009
Q: for 3.3.11  It makes sense to me that PQ could only be 0 because you project
> to a space and then the other projects to an othogonal space to the only
> shared element is 0. What def of Projection matrix do I use to show this
> though?
A:
 you use the fact that P^2 = P and P=P'; that means that P is the identity 
on any element in its own column space, say V (why?-see discussion on 
top of p.165 in Strang), while it annihilates the orthogonal complement, 
say W (i.e. W is the Null space of P).
 So, now we have P acting as the identity on V, and having W as its null
space and Q acting as the identity on W, and having V as its null space.
Then, since V _|_ W with V+W=Rn  (where P is mXn), any vector z in Rn
can be written as z = x+y uniquely, x in V, y in W. Then Px = ...,
Py = ..., Qx = ..., Qy = ...
and PQz = ..., QPz = ... for all z in Rn.
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Wed Oct  7 22:25:29 MDT 2009
Q: I'm having a hard time getting started on the first problem 2.6.6, can you
> give me a little help as to how to think about it? Is it best to draw each
> "scenario"? 
>
A: 
(a) to project every vector to the x-y plane you need a matrix that maps
   ( x  y  z) ---> ( x  y  0)
(b) to reflect through the x-y plane you need
   ( x  y  z) ---> ( x  y -z) 
(c) to rotate xy plane by 90 degrees:
   ( x  y  z) ---> ( y  -x  z)
(d) similarly to rotate xz by 90:
   ( x  y  z) ---> ( z  y  -x)
              to rotate yz by 90:
   ( x  y  z) --->  ( x -z  y)
(e) to rotate by 180:
   ( x  y  z) ---> (-x -y  z)  xy plane
   ( x  y  z) ---> (-x  y -z)  zx  ""
   ( x  y  z) ---> ( x -y -z)  yz  ""

in each case, write as column vectors and find the matrix that maps the
left to the right side.

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Fri Sep 25 00:54:34 MDT 2009
 There is a misprint in the statement of problem 21, corrected form is 
given here (see also assignment listing
(the graph on the right of p.123 (3rd Ed. p.113) has 4 nodes and 6 edges.)
2.5.21 The adjacency matrix M of a graph has M_{ij} = 1 if nodes i and j are          
      connected by an edge (otherwise M_{ij} = 0). For the graph in problem 6 
      with 6 edges and 4 nodes, write down M and also M^2. Why does (M^2)_{ij} 
      count the number of 2-step paths from node i to node j?
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Fri Sep 25 00:17:29 MDT 2009
Q: For 2.5.4 and 2.5.5 what 3 by 3 matrix A is it referring to?
A: it is the matrix A from problem 2.5.1 (same for problems 1-5)
   We derived A in class.
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 Fri Sep 11 16:44:41 MDT 2009
NOTICE CHANGE!!! problem 2.2.6 replaces 2.2.4
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    Set 3 (due 9/17): p.73, 2.1(2,7,8)
                      p.85, 2.2(5,6,7,8)  NOTICE CHANGE!!! problem 6 replaces 4
                      p.98, 2.3(2,6,8,10)
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3rd Ed.
2.1.2c: change b3 to b1 throughout.
2.2.5 4th = 2.2.6 3d + {part b: same for
                               [1 2 3][u] = [1]
                               [2 4 4][v]   [4]
                                      [w]
2.2.(6,7,8) 4th = 2.2.(7,8,9) 3rd
2.3.2 Find the largest possible number of independent vectors among
      v1=[ 1], v2=[ 1], v3=[ 1], v4=[ 0], v5=[ 0], v6=[ 0]
         [-1]     [ 0]     [ 0]     [ 1]     [ 1]     [ 0]
         [ 0]     [-1]     [ 0]     [-1]     [ 0]     [ 1]
         [ 0]     [ 0]     [-1]     [ 0]     [-1]     [-1]
      This number is the ???? of the space spanned by the v's.
2.3.6 Choose 3 independent columns of U. Then make two other choices. Do the
      same for A. You have found bases for which spaces?
U=[2 3 4 1]   and A=[2 3 4 1].
  [0 6 7 0]         [0 6 7 0]
  [0 0 0 9]         [0 0 0 9]
  [0 0 0 0]         [4 6 8 2]
2.3.8 If w1, w2, w3 are independent vectors, show that the sums v1=w2+w3,
      v2=w1+w3, and v3=w1+w2 are independent. (write c1*v1+c2*v2+c3*v3=0
      in terms of the w's. Find and solve equations for the c's.)
2.3.10 Find two independent vectors on the plane x+2*y-3*z-t=0 in R4.
      Then find 3 independent vectors. Why not 4? This plane is the
      nullspace of what matrix?
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