Chapter 3 Vector Spaces
Chapter 3
Vector Spaces
The operations of addition and scalar multiplication
are used in many contexts in mathematics. Regardless
of the context, however, these operations usually obey
the same set of algebra rules. Thus a general theory of
mathematical systems involving addition and scalar
multiplication will have application to many areas in
mathematics.
Mathematical systems of this form are called
vector spaces or linear spaces.
1 Definition and Examples
Euclidean Vector Spaces Rn
In general, scalar multiplication and addition in Rn are
defined by
nn yx
yx
yx
yx
22
11
nx
x
x
x
2
1
and
for any nRyx , and any scalar 。
The Vector Space Rm×n
Rm×n denote the set of all m×n matrices with real entries.
Definition
Let V be a set on which the operations of addition and
scalar multiplication are defined. By this we mean that,
with each pair of elements x and y in V, we can associate
a unique elements x+y that is also in V, and with each
element x in V and each scalar , we can associate a
unique element x in V. The set V together with the
operations of addition and scalar multiplication is said
to form a vector space if the following axioms are satisfied.
Vector Space Axioms
A1. x+y=y+x for any x and y in V.
A2. (x+y)+z=x+(y+z) for any x, y, z in V.
A3. There exists an element 0 in V such that x+0=x for
each x ∈V.
A4. For each x ∈V, there exists an element –x in V such
that x+(-x)=0.
A5. α(x+y)= αx+ αy for each scalar α and any x and y in V.
A6. (α+β)x=αx+βx for any scalars α and β and any x ∈V.
A7. (αβ)x=α(βx) for any scalars α and β and any x ∈V.
A8. 1·x=x for all x ∈V.
The closure properties of the two operations:
C1. If x V and ∈ α is a scalar, then αx V .∈C2. If x,y V∈ , then x+y V∈ .
Example Let W={(a,1) a real} with addition and scalar
Multiplication defined in the usual way.
Example Let S be the set of all ordered pairs of real
numbers. Define scalar multiplication and addition on
S by
)0,(),(),(
),(),(
112121
2121
yxyyxx
xxxx
We use the symbol to denote the addition operation
for this system avoid confusion with the usual addition
x+y of row vectors. Show that S, with the ordinary scalar
multiplication and addition operation ,is not a vector
space. Which of the eight axioms fail to hold?
The Vector Space C[a, b]
C[a,b] denote the set of all real-valued functions that
are defined and continuous on the closed interval
[a,b].
)())((
)()())((
xfxf
xgxfxgf
The Vector Space Pn
Pn denote the set of all polynomials of degree less
than n.
)())((
)()())((
xpxp
xqxpxqp
Theorem 3.1.1 If V is a vector space anf x is any
Element of V, then
(1) 0x=0.
(2) x+y=0 implies that y=-x.
(3) (-1)x=-x.
2 Subspaces
Example Let , S is a subset of R2.
12
2
1 2xxx
xS
Definition
If S is a nonempty subset of a vector space V, and S
satisfies the following conditions:
(1) whenever for any scalar
(2) whenever and
then S is said to be a subspace of V.
Sx Sx
Syx Sx Sy
Example
1、 Let , S is a subspace of R3. 21321 xxxxxS T
2、 Let . If either of the two
conditions in the definition fails to hold, then S will not
be a subspace.
number real a is
1x
xS
3、 Let .The set S forms a subspace
of R2×2 .
211222 aaRAS
The Nullspace of a Matrix
Let A be an m×n matrix. Let N(A) denote the set of all
solutions to the homogeneous system Ax=0. Thus
0)( AxRxAN n
The subspace N(A) is called the nullspace of A.
Example Determine N(A) if
1012
0111A
The Span of a Set of Vectors
Definition
Let v1, v2, … , vn be vectors in a vector space V. A sum of the f
orm α 1v1+ α 2v2+ ‥‥ α nvn, where α1, …, αn are scalars, is calle
d a linear combination of v1, v2, … , vn .
The set of all linear combinations of v1, v2, … , vn is called th
e span of v1, v2, … , vn . The span of v1, v2, … , vn will be den
oted by Span(v1, …, vn).
Theorem 3.2.1 If v1, v2, … , vn are elements of a vector
space V, then Span(v1, v2, … , vn ) is a subspace of V.
Spanning Set for a Vector Space
Definition
The set {v1, v2, … , vn} is a spanning set for V if and only if eve
ry vector in V can be written as a linear combination of v1, v2,
‥‥ ,vn.
Example Which of the following are spanning sets for R3?
1 2 3(a) , , , 1 2 3
(b) 1 1 1 , 1 1 0 , 1 0 0
(c) 1 0 1 , 0 1 0
(d) 1 2 4 , 2 1 3 , 4 1 1
T
T T T
T T
T T T
e e e
3 Linear Independence
Consider the following vectors in R3:
8
3
1
x,
1
3
2
x,
2
1
1
x 321
Conclusion:
(1) If v1, v2, … , vn span a vector space V and one of these
vectors can be written as a linear combination of the other
n-1 vectors, then those n-1 vectors span V.
(2) Given n vectors v1, v2, … , vn , it is possible to write one of
the vectors as a linear combination of the other n-1 vectors if
and only if there exist scalars c1, …, cn not all zero such that
c1v1+c2v2+‥‥cnvn=0
Definition
The vectors v1, v2, … , vn in a vector space V are said to be li
nearly independent if
c1v1+c2v2+‥‥+cnvn=0
implies that all the scalars c1, …, cn must equal 0.
Definition
The vectors v1, v2, … , vn in a vector space V are said to be li
nearly dependent if there exist scalars c1, …, cn not all zero
such that
c1v1+c2v2+‥‥+cnvn=0
If there are nontrivial choices of scalars for which the linear
combination c1v1+c2v2+‥‥+cnvn equals the zero vector, then
v1, v2, … , vn are linearly dependent. If the only way the linear
combination c1v1+c2v2+ +‥‥ cnvn can equal the zero vector is
for all the scalars c1, …, cn to be 0, then v1, v2, … , vn are line
arly independent.
Theorems and Examples
Example Which of the following collections of vectors are linearly independent in R3?
(a) 1 1 1 , 1 1 0 , 1 0 0
(b) 1 0 1 , 0 1 0
(c) 1 2 4 , 2 1 3 , 4 1 1
T T T
T T
T T T
Theorem 3.3.1 If x1, x2, … , xn be n vectors in Rn and let
X=(x1, …, xn). The vectors x1,x2, …, xn will be linearly
dependent if and only if X is singular.
Example Determine whether the vectors (4, 2, 3)T,
(2, 3, 1)T, and (2, -5, 3)T are linearly dependent.
Example Given
7
7
0
1
x,
2
1
3
2
x,
3
2
1
1
x 321
determine if the vectors are linearly independent.
Theorem 3.3.2 If v1, v2, … , vn be vectors in a vector
space V. A vector v in Span(v1, …, vn) can be written
uniquely as a linear combination of v1,v2, …, vn if and only
if v1,v2, …, vn are linearly independent.
4 Basis and DimensionDefinition
The vectors v1, v2, … , vn form a basis for a vector space V if
and only if
(1) v1, …, vn are linearly independent.
(2) v1, …, vn span V.
Example The standard basis for R3 is {e1, e2, e3};however,
there are many bases that we could choose for R3.
Example In R2×2, consider the set {E11, E12, E21, E22},
where
10
00,
01
00
00
10,
00
01
2221
1211
EE
EE
Theorem 3.4.1 If {v1, v2, … , vn } is a spanning set for a
vector space V, then any collection of m vectors in V,
where m>n, is linearly dependent.
Corollary 3.4.2 If {v1, v2, … , vn } and {u1, u2, … , um } are
both bases for a vector space V, then n=m.
Definition
Let V be a vector space. If V has a basis consisting of n v
ectors, we say that V has dimension n. The subspace {0} of
V is said to have dimension 0. V is said to be finite-dimensio
nal if there is a finite set of vectors that spans V; otherwise,
we say that V is infinite-dimensional.
Theorem 3.4.3 If V is a vector space of dimension n>0:
(1) Any set of n linearly independent vectors spans V.
(2) Any n vectors that span V are linearly independent.
Example Show that is a basis
for R3.
1
0
1
0
1
2
3
2
1
,,
Theorem 3.4.4 If V is a vector space of dimension
n>0,
then:
(1) No set of less than n vectors can span V.
(2) Any subset of less than n linearly independent
vectors can be extended to form a basis for V.
(3) Any spanning set containing more than n vectors
can be pared down to form a basis for V.
5 Change of Basis
Changing Coordinates in R2
x=x1e1+x2e2 the coordinate of x is (x1, x2)T
x=αy+βz the coordinate of x is (α, β)T
Changing Coordinates
Let [e1, e2] be the standard basis, [u1, u2] is another basis.
Two problems:
(1) Given a vector x=(x1, x2)T, find its coordinates with
respect to u1 and u2.
(2) Given a vector c1u1+c2u2, find its coordinates with
respect to e1 and e2.
1
1u
2
3u 21 ,
x=Uc
the matrix U is called the transition matrix from the ordered
basis [u1, u2] to the basis [e1, e2].
Example Let u1=(3,2)T, u2=(1,1)T, and x=(7,4)T. Find
the coordinates of x with respect to u1 and u2.
Example Let b1=(1,-1)T, b2=(-2,3)T. Find the transition
matrix from [e1, e2] to [b1, b2] and determine the coordinates
of x=(1,2)T with respect to [b1, b2].
Example Find the transition matrix corresponding to the
change of basis from [v1, v2] to [u1, u2], where
3
7v,
2
5v 21 and
1
1u
2
3u 21 ,
Change of Basis for a General Vector Space
Definition
Let V be a vector space and let E=[v1, v2, … , vn] be an ord
ered basis for V. If v is any element of V, then v can be writte
n in the form
v=c1v1+c2v2+‥‥+cnvn
where c1, …, cn are scalars. Thus we can associate with each
vector v a unique vector c=(c1, c2, …, cn)T in Rn. The vector
c defined in this way is called the coordinate vector of v with
respect to the ordered basis E and is denoted [v]E. The ci’s ar
e called the coordinates of v relative to E.
Example Let
E=[v1, v2, v3]=[(1, 1, 1)T, (2, 3, 2)T, (1, 5, 4)T]
F=[u1, u2, u3]=[(1, 1, 0)T, (1, 2, 0)T, (1, 2, 1)T]
Find the transition matrix from E to F. If
x=3v1+2v2-v3 and y=v1-3v2+2v3
find the coordinates of x and y with respect to the ordered
basis F.
6 Row Space and Column Space
Definition
If A is an m×n matrix, the subspace of R1×n spanned by the
row vectors of A is called the row space of A. The subspace
of Rm spanned by the column vectors of A is called the column
space of A.
Theorem 3.6.1 Two row equivalent matrices have the same
row space.
Definition
The rank of a matrix A is the dimension of the row space of A.
Example Let
741
152
321
A
Theorem 3.6.2 (Consistency Theorem for Linear Systems)
A linear system Ax=b is consistent if and only if b is in the
column space of A.
Theorem 3.6.3 Let A be an m×n matrix. The linear system
Ax=b is consistent for every b∈Rm if and only if the column
vectors of A span Rm. The system Ax=b has at most one
solution for every b∈Rm if and only if the column vectors of A
are linearly independent.
Corollary 3.6.4 An n×n matrix A is nonsingular if and only if
the column vectors of A form a basis for Rn.
Definition
The dimension of the nullspace of a matrix is called the nullity
of the matrix.
Theorem 3.6.5 ( The Rank-Nullity Theorem)
If A is an m×n matrix, then the rank of A plus the nullity of A
equals n.
Example Let
5121
0342
1121
A
Find a basis for the row space of A and a basis for N(A).
Verify that dim N(A)=n-r.
Theorem 3.6.6 If A is an m×n matrix, the dimension of the
row space of A equals the dimension of the column space of A.
Example Let
513521
43110
22031
21121
A
Find a basis for the column space of A.
Example Find the dimension of the subspace of R4
spanned by
4
5
8
3
x,
0
2
4
2
x,
2
3
5
2
x,
0
1
2
1
x 4321