# Math 2270-1
# Project 5
# 
# In this project you will use Maple to solve some of the problems we
# have been discussing in class: components with respect to a basis,
# transition matrix from one basis to another, orthonormal basis,
# Gram-Schmidt process.  If you want to use version 4 of maple, remember
# to type
# 
# xmapleV4 &
# 
# and then hit return.  Make sure that "V" is capitalized.  You can also
# get version 4 from the menu you get from the middle mouse button.
# 
# 
# Problem 1: Components and Transition Matrices.
# 
# This is essentially problem 14 of section 4.7.  We first define two
# basis S and T of R^3 to be the columns of the following matrices,
# called MS and MT:
> with(linalg):
> MS:=matrix([[1,-1,0],[0,0,1],[1,0,2]]);

                               [1    -1    0]
                               [            ]
                         MS := [0     0    1]
                               [            ]
                               [1     0    2]

> MT:=matrix([[-1,1,0],[1,2,1],[0,-1,0]]);

                              [-1     1    0]
                              [             ]
                        MT := [ 1     2    1]
                              [             ]
                              [ 0    -1    0]

# Let v and w be the following vectors:
# 
> v:=vector([1,3,8]);w:=vector([-1,8,2]);

                            v := [1, 3, 8]


                           w := [-1, 8, 2]

# Problem 1a:  Find the coordinate vectors of v and w with respect to
# the basis T.  (Hint: you could use "linsolve").
# 
# Problem 1b:  Find the transition matrix P(S<-T) from basis T to basis
# S.  You could  use the procedure described on p. 261: make a matrix (
# MS | MT) and take its reduced row echelon form.   You could  then use
# the command "delcols" (look it up) to find the submatrix that computes
# P(S<-T)
#  
# 
# Problem 1c: Find the coordinates of v and w in the basis S by using
# the transition matrix P(S<-T).
# 
# 
# Problem 1d:  Evaluate the components of v and w in the basis S
# directly, by the same method you used in problem 1a to find the
# components in the basis T.  Compare the two answers.
# 
# Problem 1e: Use the same method as in problem 1b to find the
# transition matrix P(T<-S) from the S basis to the T basis.  Then
# verify that P(T<-S) is the inverse of P(S<-T).
# 
# Problem 1f: Find the coordinate vectors of v and w with respect to the
# basis T by using their components with respect to the basis S and the
# transition matrix P(T<-S).  Compare your answer with problem 1a.
# 
# 
# Problem 2: Orthogonal complements.
# This is problem 5 of section 4.8.  Let A be the matrix 
> A:=matrix([[2,-1,3,4],[0,-3,7,2],[1,1,-2,3],[1,4,-9,5]]);

                           [2    -1     3    4]
                           [                  ]
                           [0    -3     7    2]
                      A := [                  ]
                           [1     1    -2    3]
                           [                  ]
                           [1     4    -9    5]

# Problem 2a: Find bases for the null space and the row space of A, and
# use these bases to verify that the null space is the orthogonal
# complement of the row space.  You could verify this by hand, or by
# using Maple.
# 
# Problem 2b: Find bases for the column space of A and the null space of
# the transpose of A, and verify that these two spaces are orthogonal
# complements of each other.
# 
# Problem 2c: Use the Gram-Schmidt process to find an orthonormal basis
# for the column space of A.  Look up how to use the Gram-Schmidt
# process.  You'll find that it only gives you an orthogonal basis, you
# would then have to use "normalize" (look up) to get an orthonormal
# basis.