Homework 01, 05/22
   Read Section 1.1 and 1.2
   Section 1.1: #14(c,d)
   Section 1.2: #4, 6(a-d), 7, 23(a,e), 24
   Additional problems: 
   1. Find two different vectors of length 5 that is perpendicuar to 
             v=(1,3,-2)
   2. Simplify, using Theorem 1.1 and 1.3
       (i)     [4(3u-5v)+7u].(2u+3v)
       (ii)     5(2u-5v)-3[4u-5(v+2u)]
       (iii)   (||u|| + ||v||)^2 - (u+v).(u-v)
   3. True or false, give a short explanation or example, 
      where a, b, s are scalars and a, b, c, 0 are vectors
       (i)    if ab = 0 then either a = 0 or b = 0
       (ii)   if a.b = 0 then either a = 0 or b = 0
       (iii)  if sa = b then a = (1/s)b
       (iv)   if a.b = 5 then b = 5 / a
       (v)    if a - b = c then a = b + c
       (vi)   a(bc) = (ab)c
       (vii)  a.(b.c) = (a.b).c

Homework 02, 05/24
   Read Section 2.3 and 2.4
   Section 2.3: #2, 4, 7, 8, 10, 14
   Section 2.4: # 8, 9, 10, 20(a,c,e), 21(a)
   Additional problems: Use properties of matrix operations to prove the following.
   1. A matrix A is called symmetric if A = At. Show that if A is symmetric
      then so is its transpose.
   2. Let A be an nxn matrix. Show that AtA is symmetric.
   3.(Extra credit) Let u be a column vector of dimension n. Further assume that u is a unit
      vector, i.e. ||u||=1. The Householder matrix is defined by
                       H = I - 2uut
        (i) Show that H is symmetric
      (ii) Show that HH = I

Homework 03, 05/31
   Read Section 2.4, 3.2
   Section 2.4: #24(a,c,e)
   Section 3.2: #2, 5, 8, 11, 14
   Additional problems: 
   1. Let u be a column vector of dimension n. Further assume that u is a unit
      vector, i.e. ||u||=1. The Householder matrix is defined by
                       H = I - 2uut

        (i) Let u=(5,2,-3)t, find the corresponding Householder matrix
      (i) prove that H=H-1.
      (ii) prove that Ht=H-1.
      (iii) prove that for all n-vector x, Ax - x is parallel to u.
   2. Let A be an nxn matrix such that A3=O. Prove that
            (I-A)-1=I+A+A2.
   3. Let A, B be two nxn invertible matrices such that AB=BA. 
      Prove that A-1B-1=B-1A-1  .
   4. True or false problems. For ture statements, give a short explanation.
      For false statements, give an example.
      (i) Every matrix has inverse.
      (ii) Every number has an inverse.
      (iii) AB=0 then A=0 or B=0.
      (iv) Linear system Ax=b always has a solution.
      (v) Linear system Ax=0 always has a solution.
      (vi) If both A and B are invertible, so is AB.
      (vii) Linear system Ax=b always has a solution when A is invertible

Homework 04, 06/05
   Theoretical Exercises I: #1-3, 5, 7, 9, 11, 12, 15, 16, 22

Homework 05, 06/07
   Section 4.1, #12
   Section 4.2, #8, 9, 10
   Section 4.1, #1, 2, 5, 13, 14, 26, 27
   Additional problem:
   Show that {all vectors parallel to (1,2,-1)} is a subspace
of R3

Homework 06, 06/012
   Section 4.3, #30, 31
   Additional problem:
   Theoretical Exercise II: #1-4, 7-8, 10-12
   Extra credit: #5-6, 20

Homework 07, 06/14
   Section 4.4, #3,4,11,12,14,16
   Theoretical Excercise II: #9,10, Extra credit: #14, 16, 19

Homework 08, 06/19
   Section 5.2, #3, 4, 5(a), 16
   Section 5.3, #2, 3, 4
   Extra credit: Section 5.2, #11