# Homework #4
# 
# Due: April 18th, @ 23:45
# 
# Name: KEY
# 
# Grade:  39/39
# 
# Do the following problems showing all of your work.  You do not have
# to use
# Maple commands for all your computations, but you are encoraged to
# explore
# the linalg package and use it for your computations.

> 
> with(linalg):
Warning, the protected names norm and trace have been redefined and
unprotected

# Problem 1:
# 
# Given the matrix:
# 
#  A = matrix([[1, 3, 2, 0], [4, -1, 20, 4], [7, 8, 26, 4]])
# 
# and vector
# 
#  b = [-12, 28, -8]
> 
# a) Find a basis for the null space of A.
> 
> A := matrix([[1, 3, 2, 0], [4, -1, 20, 4], [7, 8, 26, 4]]);

                           [1     3     2    0]
                           [                  ]
                      A := [4    -1    20    4]
                           [                  ]
                           [7     8    26    4]

> nullspace(A);

                  {[-2, 0, 1, -3], [-3, 1, 0, 13/4]}

> # 3/3
# b) Find a basis for the column space of A.
> 
> basis(A, colspace);

                       [[1, 4, 7], [3, -1, 8]]

> 
> # 3/3
# c) Find a basis for the row space of A.
> 
> basis(A, rowspace);

                    [[1, 3, 2, 0], [4, -1, 20, 4]]

> 
> # 3/3
# d) Find one solution to A x = b.
> b := vector([-12, 28, -8]); t := seq(0,i=1..3);

                          b := [-12, 28, -8]


                             t := 0, 0, 0

> linsolve(A,b,'foo',t);

                           [-12, 0, 0, 19]

> # 3/3
# e) Find an explicit representation for /all/ solutions to A x = b
# using the
#    basis for the null space.
> # Every solution must be in the following form:
> x0 := vector([-12, 0, 0, 19]) + a[1]*vector([-2, 0, 1, -3]) + a[2] *
> vector([-3, 1, 0, 13/4]);

 x0 := [-12, 0, 0, 19] + a[1] [-2, 0, 1, -3] + a[2] [-3, 1, 0, 13/4]

> evalm(A&*x0-b);

                              [0, 0, 0]

> 
> # 3/3
# Problem 2:
# 
# Let u = [3, 6, 7, 8, 0], v = [6, 7, 8, 0, 3], w = [-3,8,-7,5,6]. 
# 
# Compute the following:
> 
# a) The scalar products: u^T v, v^T u, w^T v.
> 
> u := vector([3, 6, 7, 8, 0]); v := vector([6, 7, 8, 0, 3]); w :=
> vector([-3,8,-7,5,6]);

                         u := [3, 6, 7, 8, 0]


                         v := [6, 7, 8, 0, 3]


                        w := [-3, 8, -7, 5, 6]

> dotprod(u,v);dotprod(u,w);dotprod(w,v);

                                 116


                                  30


                                  0

> # 3/3
# b) Are u and v orthogonal? Are v and w? Are u and w?
> 
#  u and v are not orthogonal.
# u and w are not orthogonal
# v and w are orthogonal
# c) The Euclidean norm of u,v, and u-v: ||u||, ||v||, ||u-v||.
> 
> norm(u,2);

                              sqrt(158)

> norm(v,2);

                              sqrt(158)

> norm(u-v,2);

                              2 sqrt(21)

> # 3/3
# d) The angle between u and v.
> 
> arccos(dotprod(u,v)/(norm(u,2)*norm(v,2)));

                                     58
                              arccos(--)
                                     79

> evalf(%); # in radians

                             .7463422759

> evalf(% *180/Pi); # in degrees

                             42.76226248

> # 3/3
# e) The projection of u onto v.
> proju_v := evalm(dotprod(u,v)/dotprod(v,v) * v);

                             [348  406  464     174]
                  proju_v := [---, ---, ---, 0, ---]
                             [79   79   79      79 ]

> # 3/3
> 
# f) The projection of u onto span(w,v).
> 
> # Note w is orthogonal to v so we can just add the projections of u to
> w and to v.
> proju_vw := evalm(proju_v + dotprod(u,w)/dotprod(w,w) *w);

                         [18858  31086  22774  50  15354]
             proju_vw := [-----, -----, -----, --, -----]
                         [4819   4819   4819   61  4819 ]

> # 3/3
# g) The orthogonal compliment of the space V = span(u,v,w).
> 
> V := matrix([u,v,w]);

                         [ 3    6     7    8    0]
                         [                       ]
                    V := [ 6    7     8    0    3]
                         [                       ]
                         [-3    8    -7    5    6]

> nullspace(V);

           [73                    ]
          {[--, 0, -15/4, 1, -13/6], [8/3, 1, -2, 0, -7/3]}
           [12                    ]

> # 3/3
# Problem 3 (Theory):
> 
# a) Show that if for vectors u,v, and w, that if w is orthogonal to
# both u and v then w is orthogonal to every vector in span(u,v).
> 
#  If <w,u> = 0 and <w,v> = 0, and x is in span(u,v) then x can be
# written as x = a*u + b*v.  So, <w,x> = <w, a*u+b*v> = a*<w,u> +
# b*<w,v> = 0.  Thus w is orthogonal to every element in span(u,v).
#  
> 3/3
# b) Show that if w in a vector space V (a subspace of R^n) and w is
# also in the othogonal compliment of V, then w=0. 
> 
#  If w is in both V and its orthogonal compliment, then <w,w> = 0.  If
# w = [w1, .... , wn], then this means that w1^2 + ... + wn^2 = 0.  But
# this is a sum of non-negative numbers, hence they all must be zero. 
# So w = [0, ...., 0].
> 
#  
> 3/3