RANK and
NULLITY
1. Row and Column Spaces Have Equal Dimensions THEOREM 5.6.1 If A is any matrix, then the row space and column space of A have the same dimension dim(row space of A) = dim(column space of A) DEFINITION The common dimension of the r

LINEAR INDEPENDENCE DEFINITION If S = {v1, v2,., vr,} is a nonempty set of vectors, then the vector equation k1v1 + k2v2+ . + krvr = 0 has at least one solution, namely k1 = 0, k2 = 0, . kr = 0 If this is the only solution, then S is called a linearl

2. Calculating
THEOREM
projwu
and
u projwu.
If u and w are vectors in 2-spase or 3-space and if w 0, then
projw u =
ui w uiw w= w (vector component of u along w ) 2 wiw w ui w w (vector component of u orthogonal to w ) wiw
u - projw u = u -

VECTOR SPACES Many concepts concerning vectors in Rn can be extended to other mathematical systems. We can think of a vector space in general, as a collection of objects that behave as vectors do in Rn. The objects of such a set are called vectors. 1

Linear and Nonlinear equations
D.J.Jeffrey Dept Applied Maths, UWO September 25, 2006
1
Introduction
The first step in solving a set of equations is identification. Equations can be classified into various types, and once we have identified a set

Exercises for AM025
D.J. Jeffrey Department of Applied Mathematics U. Western Ontario July 19, 2006
1
Linear and nonlinear
1. The set of two equations ap2 x - bpqy = 8 and a + pqx + y = 1 is (a) nonlinear in {x, y} and nonlinear in {a, b}. (b) line

1
INNER PRODUCT SPACES Definition Two vectors u and v in an inner product space are called orthogonal if an inner product is equal to zero: < u, v > = 0. A Geometric Link between Nullspace and Row Space of a Matrix Theorem 6.2.6 If A is an m n matr

Some Engineering Applications
D.J. Jeffrey January 11, 2006
These notes supplement those in your textbook.
1
Flow of water in a pipe network
21 c 15 ' A i1 13 E B c 12 i4 D T 10 i3 8 T C i2 E 25 ' 16
Consider the network of pipes shown in figure 1

THE UNIVERSITY OF WESTERN ONTARIO DEPARTMENT OF APPLIED MATHEMATICS Applied Mathematics 025a Third Tutorial Test, November 22, 2007 Time: 60 minutes The test is closed book! You are allowed to use only simple calculators.
Section 002, Professor Kiri

BASIS and DIMENTION Nonrectangular Coordinate Systems In plane analytic geometry we associate a point P in the plane with a pair of coordinates (a, b) by projecting P onto a pair of perpendicular coordinate axes (Fig. 1a). Each point in the plane is

CHEM 024b: Introductory University Chemistry for Engineers Midterm Examination WSC 55 24 pm, Saturday February 17th, 2007 Important information and instructions. Please read carefully. (1) This exam is made up of 25 multiple choice questions and 5 sh