1
NOTES FOR
LINEAR ALGEBRA 133
William J. Anderson
McGill University
These are not ocial notes for Math 133. They are intended for Andersons section 4, and are
identical to the notes projected in class.
2
Contents
1 Linear Equations and Matrices.
1.1 Line
MATHEMATICS 133 ASSIGNMENT 4, Due Feb 12 in Class
1 4
1
7 9
0
.
1. Let A =
10
3 3
9
1
2
(a) Find the reduced row echelon form of A. [3p].
(b) Compute Null A = Ker A, Im A = col(A) and row(A). [6p].
(c) Find bases for all these spaces. [4p].
(d) Find
MATHEMATICS 133 ASSIGNMENT 6, Due Mar 17 in Class
1. Let X = [1, 2, 3]T , Y = [2, 1, 2]T and Z = [3, 3, 2]T . Compute [2p
each]:
(a) sin , where is the angle between X and Y ,
(b) cos , where is the angle between X and Z,
(c) Z (X Y ),
(d) Y (Y Z),
(e) X
Students Last Name : I Student Number :
Other name(s) :
FACULTY OF SCIENCE FINAL EXAMINATION
MATHEMATICS 133 VECTORS, MATRICES AND GEOMETRY
Examiner: Dr. I. Klemes Date: Monday, 16 April 2007.
Associate Examiner: Dr. B. Charbonneau Time: 9:00 am 12:
PREP 101 - MATH133
Chapter One - solutions
1. Solution
a) u v = u v = 0
1
2
(3k ) + k (6) 3 (4) = 0 = 3k =
3
3
b) |u| = 1 =
1
9
+
k2
9
+
4
9
8
3
= k =
8
9
= 1 = k 2 = 4 = k = 2 or k = 2
2. Solution
a) AB = (2, 2, 4).
b) Midpoint of AB is (2, 6, 1).
c) Cal
MATH 133 - Formula Sheet
Denition(Norm of a vector) v1 v2 If v = . in Rn the norm of v is given by |v| = . . vn Denition(Dot Product) u1 u2 If u = . and v = . . un
2 2 2 v1 + v2 + + vn
v1 v2 . . . vn
then the dot product of u and v is dened by
u v = u1