4 Vector Spaces
4.1
VECTOR SPACES AND
SUBSPACES
© 2012 Pearson Education, Inc.
Topics
Vectors in the Plane and in 3D-Space
Vectors in the Plane and in 3-Space
Defn - A vector in the plane is a 2 x 1 matrix v
where x and y are real numbers
Notation - In print, use bold letters for vectors.
When writing by hand use v
Defn - The numbers x and y in the definition of a
vector are called the components of the vector v.
Defn - Two vectors v
are equal iff
x1
and u
y1
x1 = x2
and
x2
y2
y1 = y2
x
y
Vectors in the Plane and in 3-Space
x has several geometric
y
A two-dimensional vector v
interpretations
1) A point ( x,y ) in the plane
2) A directed line segment from the origin to the point ( x,y )
3) A directed line segment from the point ( x1,y1 ) to the
point ( x2,y2 ). Then x = x2 x1, y = y2 y1
For applications to geometry and physics, some of the most
important operations are
1) Vector addition
2) Scalar multiplication
3) Vector subtraction
Vectors in the Plane and in 3-Space
Vector Addition
u1
Defn - Let u
and v
u2
Geometric interpretation
v1
. Define u v
v2
u1 v1
u2 v2
Vectors in the Plane and in 3-Space
Vector Addition
u + v can be determined geometrically by adding
vectors head to tail
y
If u and v have the same
v
length (to be defined later),
then u + v lies along the
u v
u
bisector of the angle
between u and v
v
x
Vectors in the Plane and in 3-Space
Scalar Multiplication
Defn - Let c be a scalar (e.g. a real number) and u
be a vector. The scalar multiple c u of u by c is
defined as the vector cu1
cu2
u1
u2
Vectors in the Plane and in 3-Space
Vector Subtraction
Define u v as u + ( 1) v
Geometric relationship among u,v, u + v and u
v
Vectors in the Plane and in 3-Space
Parametric Description of a Line
Let P1 and P2 be points. The line passing through
them can be described as
1 t P1 tP2
P P1 t P2 P1
P2
P2 P1
t
t
0
t 1
t 1
0 t 1
P1
0
t
Vectors in the Plane and in 3-Space
x
Three-Dimensional Vectors
A three-dimensional vector is a 3 x 1 matrix y
z
where x, y and z are real numbers
Vector addition, scalar multiplication and vector
subtraction are defined analogously to two
dimensions
Vectors in the Plane and in 3-Space
Basic Properties of Vectors in R2 or R3
Theorem - If u, v and w are vectors in R2 or R3
and c and d are real scalars, then
a)u + v = v + u
b) u + (v + w) = (u + v) + w
c)u + 0 = 0 + u = u
d) u + ( u) = 0
e)c (u + v) = c u + c v
f) (c + d) u = c u + d u
g) c (d u) = (c d ) u
h) 1 u = u
VECTOR SPACES AND SUBSPACES
Definition:
A vector space is a nonempty set V of objects,
called vectors,
on which are defined two operations,
called addition
and multiplication by scalars
(real numbers),
subject to the ten axioms (or rules) listed below.
The axioms must hold for all vectors u, v, and w in V
and for all scalars c and d.
Slide 4.1- 12
Vector Spaces
Axioms
1) If u, v Î∈ V, then u v V
2) u v w
u v w for all u, v, w Î∈ V
3) $∃ a unique 0 Î∈ V such that 0 u u 0 u
4) "∀ u Î∈ V $∃ a unique v Î∈ V such that u v v
5) u v v u "∀ u, v Î∈ V
6) "∀ u Î∈ V and "∀ real number c, c u Î∈ V
c
7) "∀ real numbers c and "∀ u, v Î∈ V, c u v
8) "∀ real numbers c and d, "∀ u Î∈ V, c d u c
9) "∀ real numbers c and d, "∀ u Î∈ V, c d u cd
10) 1 u u "∀ u Î∈ V
u V
u 0
u
u
u
c v
d u
Vector Spaces
Comments
V is called a real vector space because scalars c
and d are real numbers, not because of any
characteristics of the elements of V, e.g. if the
elements of V involve complex numbers, but c and
d are real, then V is a real vector space
The identity element 0 for
does not need to
have anything to do with the number zero
There is nothing in the definition that requires the
elements of V to be columns of numbers
Vector Spaces
Comments
To show that V is a vector space, we have to prove
properties 1 through 10. So, the verification
involves ten small proofs
To show that V is not a vector space, all that we
have to do is show that a single property does not
hold
VECTOR SPACES AND SUBSPACES
For each u in V and scalar c,
0u 0
c0 0
u ( 1)u
© 2012 Pearson Education, Inc.
Slide 4.1- 16
SUBSPACES
Definition: A subspace of a vector space V is a
subset H of V that has three properties:
a. The zero vector of V is in H.
b. H is closed under vector addition. That is, for
each u and v in H, the sum u v is in H.
c. H is closed under multiplication by scalars.
That is, for each u in H and each scalar c, the
vector cu is in H.
© 2012 Pearson Education, Inc.
Slide 4.1- 17
SUBSPACES
Properties (a), (b), and (c) guarantee that a subspace
H of V is itself a vector space, under the vector
space operations already defined in V.
Every subspace is a vector space.
Conversely, every vector space is a subspace (of
itself and possibly of other larger spaces).
© 2012 Pearson Education, Inc.
Slide 4.1- 18
Vector Spaces: Examples – 2D and 3D vectors
Example
The set of two-dimensional vectors, as defined
earlier, forms a vector space. The verification of
the properties follows from the properties of
matrices
Example
The set of three-dimensional vectors, as defined
earlier, forms a vector space. The verification of
the properties follows from the properties of
matrices
Vector Spaces: Examples – nD vectors & matrices
x1
Example
The set Rn of all n x 1 matrices with real entries x2
forms a vector space with as matrix addition
and as scalar multiplication
xn
Example
The set mRn of all real m x n matrices forms a
vector space with as matrix addition and as
scalar multiplication
Vector Spaces: Examples - Polynomials
Example
Consider polynomials in a variable t
pt
a0t n a1t n 1
an 1t an
b0t n b1t n 1
bn 1t bn
where a0, a1, …, an are real numbers. If a0 ≠ 0, the degree of p(t) is n. The zero polynomial
0t n 0t n 1
0t 0
has no
degree. Polynomials of degree 0 are constants. Let
Pn be the set of all polynomials of degree ≤ n
together with the zero polynomial. Let p(t) be as
above and let
qt
Vector Spaces : Examples - Polynomials
Example (continued)
Define p t q t as
pt
qt
a0 b0 t n
a1 b1 t n 1
an 1 bn 1 t
For a real number c, define c p t
c pt
ca0 t n
ca1 t n 1
Pn forms a vector space
can 1 t
as
can
an bn
Vector Spaces : Examples – Continuous fonctions
Example
Let V be the set of all real valued continuous
functions on the closed interval [ 0,1 ].
For f, g Î∈ V, define f g as
f
g t
f t
g t
t
0,1
For f Î∈ V and a real number c, define c f as
c f t
cf t
V is a vector space.
It is commonly called C [ 0,1 ].
t
0,1
Vector Spaces
Theorem - If V is a vector space, then
a) 0 u 0 for every u in V
b) c 0 0 for every scalar c
c) If c u 0 , then either c = 0 or u = 0
d) 1 u u for every u in V
A SUBSPACE SPANNED BY A SET
The set consisting of only the zero vector in a vector
space V is a subspace of V, called the zero subspace
and written as {0}.
As the term linear combination refers to any sum of
scalar multiples of vectors, and Span {v1,…,vp}
denotes the set of all vectors that can be written as
linear combinations of v1,…,vp.
© 2012 Pearson Education, Inc.
Slide 4.1- 25
A SUBSPACE SPANNED BY A SET
Example 2: Given v1 and v2 in a vector space V, let
H Span{v1 ,v 2 } . Show that H is a subspace of V.
Solution: The zero vector is in H, since 0 0v1 0v 2 .
To show that H is closed under vector addition, take
two arbitrary vectors in H, say,
u s1v1 s2 v 2 and w t1v1 t2 v2.
By Axioms 2, 3, and 8 for the vector space V,
u w
( s1v1 s2 v 2 ) (t1v1 t2 v 2 )
( s1 t1 )v1 ( s2 t2 )v 2
© 2012 Pearson Education, Inc.
Slide 4.1- 26
A SUBSPACE SPANNED BY A SET
So u
w is in H.
Furthermore, if c is any scalar, then by Axioms 7 and 9,
cu c( s1v1 s2 v2 ) (cs1 )v1 (cs2 )v 2
which shows that cu is in H and H is closed under scalar
multiplication.
Thus H is a subspace of V.
© 2012 Pearson Education, Inc.
Slide 4.1- 27
A SUBSPACE SPANNED BY A SET
Theorem 1: If v1,…,vp are in a vector space V, then
Span {v1,…,vp} is a subspace of V.
We call Span {v1,…,vp} the subspace spanned (or
generated) by {v1,…,vp}.
Give any subspace H of V, a spanning (or
generating) set for H is a set {v1,…,vp} in H such
that
H Span{v1 ,...v p }.
© 2012 Pearson Education, Inc.
Slide 4.1- 28