1. Vector addition is
COMMUTATIVE
:
a+ b= b+ a
2. Vector addition is
ASSOCIATIVE
:
a+ (b+ c) = (a+ b) + c
3. There is a zero vector 0such that:
a+ 0= a
4. Every vector ahas a negative –asuch that:
a+ (–a) = 0
5. For every vector awe have:
1.a= a
6. If ris a scalar and aand bare vectors, then:
r(a+ b) = ra+ rb
7. If rand sare scalars and ais a vector, then:
(r+ s)a= ra+ sa
8. If rand sare scalars and ais a vector, then:
(rs)a= r(sa)
A vector space is any set Vfor which it is possible to
define a notion of addition (that is, a rule that com-
bines two elements of the set Vto produce a new ele-
ment of V) and scalar multiplication (that is, a means
to multiply elements of Vby numbers) so that the
above eight rules hold. Certainly the set of all two-
dimensional vectors forms a vector space, as does the
set of all three-dimensional vectors, but examples of
vector spaces need not be of this type. For example, let
Vbe the set of all functions from the set of all real
numbers to the set of real numbers. One can add any
two functions fand g:
(f+ g)(x) = f(x) + g(x)
and multiply functions by numbers:
(rf)(x) = rf(x)
One checks that all eight rules above hold (here 0
is the function that takes the constant value zero), mak-
ing Va vector space. Also, the set of all 3×3 matrices is
a vector space (one can add two matrices and multiply
matrices by scalars), as is the set of all
COMPLEX NUM
-
BERS
. (One can add two complex numbers and one can
multiply complex numbers by real numbers.)
Properties of vectors and their algebraic manipula-
tions have been studied extensively by scholars for cen-
turies. That mathematicians have isolated the eight key
properties that make vectors work the way they do
allows one to immediately apply all that is known
about vectors to any system, no matter how abstract it
may be, that satisfies these eight basic rules. For exam-
ple, mathematicians have proved that every vector space
must have a
BASIS
. Consequently, there must be a basis
for the set of all functions and for the set of all 3×3
matrices. Identifying one possible basis for the set of all
functions leads to a study of F
OURIER SERIES
.
See also
GROUP
;
LINEARLY DEPENDENT AND INDE
-
PENDENT
.
velocity The study of motion examines three funda-
mental notions: distance, velocity, and acceleration.
The distance traveled by a moving object (also
called its displacement) is the total length of the path it
moved along. If the object travels along a straight-line
path, then its motion is said to be rectilinear. (Motion
that is not rectilinear is called curvilinear.) If, in addi-
tion to being rectilinear, an object travels equal dis-
tances Din equal periods of time T, then its motion is
said to be uniform. This is the easiest type of motion to
analyze. In this setting, the ratio is constant and
is called the (uniform) velocity vof the object, v= ,
and the picture of a velocity-vs.-time graph is a hori-
zontal line at height v. The formula D= v ×T, coinci-
dentally, is the equation for the area of the rectangle of
width Tand height v. This shows, in this simple sce-
nario at least, that displacement equals the area under
the velocity graph.
If, in rectilinear motion, the speed of an object
changes with time, as when a car accelerates from rest to
highway speed, then the analysis of velocity and displace-
ment is more complicated. If f(t) denotes the position
D
–
T
D
–
T
522 velocity