Homework #4

Due: April 18th, @ 23:45

Name:


Collaborators:



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.


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.

b) Find a basis for the column space of A.

c) Find a basis for the row space of A.

d) Find one solution to A x = b.

e) Find an explicit representation for /all/ solutions to A x = b using the
   basis for the null space.


Problem 2:

Let u = [3, 6, 7, 8, 0]^T, v = [6, 7, 8, 0, 3]^T, w = [-3,8,-7,5,6]^T. 

Compute the following:

a) The scalar products: u^T v, v^T u, w^T v.

b) Are u and v orthogonal? Are v and w? Are u and w?

c) The Euclidean norm of u,v, and u-v: ||u||, ||v||, ||u-v||.

d) The angle between u and v.

e) The projection of u onto v.

f) The projection of u onto span(v,w).

g) The orthogonal compliment of the space V = span(u,v,w).

Problem 3:

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).

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.