DEFINITION:
A vector space is a nonempty set V of
objects, called vectors, on which are dened two operations, called addition and
multiplication by scalars (real numbers),
subject to the following 10 axioms (or
rules):
1. The sum of u and v , denoted by
u
DEFINITION:
Let V be a vector space and B be a basis
of V. The dimension of V is a number
of vectors in B.
EXAMPLE:
Since
1
0
0
0
1
0
e1 = . , e2 = . , . . . , en = .
.
.
.
0
0
1
is the basis for Rn, we get dim Rn = n.
EXAMPLE:
Find the dimension of t
VECTOR SPACES:
x1
x2
1. Rn = . : x1, . . . , xn R
.
xn
2. The set Pn of polynomials of degree
at most n:
p(t) = antn + . . . + a2t2 + a1t + a0
where the coecients an, . . . , a0 and the
variable t are real numbers.
3. The set of all real-valued functi
Lecture 4
Vectors
Objectives:
Contravariant and covariant vectors, one-forms.
Reading: Schutz chapter 3; Hobson chapter 3
4.1
Scalar or dot product
We have had
V V = V V .
If A and B are four-vectors then V with components
V = A + B ,
is also a four-vect
DEFINITION:
If u and v are vectors in Rn, then uT v is
called the inner product (or dot product)
of u and v and written as
uv
REMARK:
In other words, if
u1
.
u = . and
un
v1
.
v = . ,
vn
then
v1
.
u v = uT v = [u1 . . . un] .
vn
= u1v1 + . . . + unvn.
DE
PROBLEM:
Let T : R2 R2 be a linear operator
such that
3 2
T () = Ax, A =
x
.
10
Find all nonzero vectors x R2 and all
scalars such that
T () = x.
x
SOLUTION:
Suppose there is a vector x R2 and a
scalar such that
T () = x.
x
Since T () = Ax, we rewrite thi
PROBLEM:
Let E = cfw_e1, e2 be the standard basis
for vector spaces V and W. Let also
T :V W
be a linear transformation such that
T (1) = 31 22,
e
e
e
T (2) = 41 + 72.
e
e
e
Find the matrix M for the linear transformation T relative to the basis E.
SOLUTI
PROBLEM:
Let T : R2 R2 be a linear operator
such that
74
T () = Ax, A =
x
.
3 1
Let also
x1 =
2
,
3
x2 =
2
,
1
x3 =
Find T (1), T (2), and T (3).
x
x
x
1
.
1
SOLUTION:
We have
T (1) =
x
74
3 1
2
3
=
2
3
T (2) =
x
74
3 1
2
1
=
10
5
=
11
4
T (3) =
x
74
3 1
DEFINITION:
Let H be a subspace of a vector space
V. A set of vectors
B = cfw_1, . . . , p
b
b
in V is a basis for H if
(a) B is a linearly independent set;
(b) H = Span cfw_1, . . . , p.
b
b
REMARK:
In other words, a basis for H is a minimal number of ve
PROBLEM:
Find the angle between vectors
6
5
u = 3 and v = 9 .
1
3
THE INNER PRODUCT
DEFINITION:
If u and v are vectors in Rn, then uT v is
called the inner product (or dot product)
of u and v and written as
uv
REMARK:
In other words, if
u1
.
u = . and
un
3.1.1. The graphical method
Assume two vectors and are defined. If is added to , a third vector is created
(see Figure 3.1).
Figure 3.1. Commutative Law of Vector Addition.
There are however two ways of combining the vectors and (see Figure 3.1).
Inspecti
THEOREM:
Let S = cfw_u1, . . . , up be an orthogonal
basis for a subspace W of Rn. For each
y in W the weights in the linear combi
nation
y = c 1 u1 + . . . + c p up
are given by
y uj
cj =
uj u j
(j = 1, . . . , p).
PROBLEM:
Let S = cfw_u1, u2, u3, where
PROBLEM:
Let u and y be nonzero vectors in Rn.
Find vectors y and z such that
y = y + z,
where y is a multiple of u and z is or
thogonal to u.
DEFINITION:
The vector
y=
yu
u
uu
is called the orthogonal projection of y
onto u and denoted by
projuy .
The ve
3.1.2. The analytical method
To demonstrate the use of the analytical method of vector addition, we limit ourselves
to 2 dimensions. Define a coordinate system with an x-axis and y-axis (see Figure
3.5). We can always find 2 vectors, and , whose vector su
3.2. Multiplying Vectors - Multiplying a Vector by a Scalar
The product of a vector and a scalar s is a new vector, whose direction is the same as
that of if s is positive or opposite to that direction if s is negative (see Figure 3.8).
The magnitude of t