Chapter 3
Vector Spaces
Definition and Examples
Vectors is R2
• The sum.
a
b
c
d
a+c
b+d
• Scalar multiplication.
• The length.
Area of a parallelogram
A
B
A=B
A
B
A=B
• If v =
(
a
b
)
, w =
(
c
d
)
then the area of the parallelogram
determined by v,w does not change if we add a scalar mul-
tiple of v to w.
•∣
∣
∣
∣
∣
a b
c d
∣
∣
∣
∣
∣
=
∣
∣
∣
∣
∣
a b
c + αa d + αb
∣
∣
∣
∣
∣
• Area of the parallelogram determined by
(
x
0
)
,
(
0y
)
is xy
and we can reduce a matrix to diagonal using row operations
of type 1.
• Area of the parallelogram determined by v,w is
∣
∣
∣
∣
∣
a b
c d
∣
∣
∣
∣
∣
Examples of vector spaces
• R, R2, Rn
• Rm×n
Axioms
Definition 1 (Real Vector Space) A set V with the opera-
tions of addition + and scalar multiplication · is called a vector
space if for every x,y ∈ V and α ∈ R, x + y ∈ V , αx ∈ V and the
following axioms are satisfied.
A1. x + y = y + x.
A2. (x + y) + z = x + (y + z).
A3. There is 0 ∈ V such that x + 0 = x.
A4. For every x ∈ V there is −x ∈ V such that x + −x = 0.
A5. For α ∈ R and x, y ∈ V , α(x + y) = αx + αy.
A6. For α, β ∈ R and x ∈ V , (α + β)x = αx + βx.
A7. For α, β ∈ R and x ∈ V , (αβ)x = α(βx).
A8. For x ∈ V , 1x = x.
Remark: Instead of R a different set of scalars can be used.
Example 1 • C[a, b].
• C over R.
• Pn, the space of polynomials of degree at most n.
• Q(√
2). The set of scalars is Q.
• The set of bounded functions f : [a, b] → R.
Subspaces
Let (V,+, ·) be a vector space and let ∅ 6= W ⊆ V . If (W,+, ·)is a vector space then it is called a subspace of V . Equivalently:
W is called a subspace of V if
• αv ∈ W for every v ∈ W and any scalar α.
• v + w ∈ W for every v,w ∈ W .
Example 2 • W = {(x1, x2, x3)T |x1 + x2 + x3 = 0} is a sub-
space of V .
• Let W be the set of functions which have a continuous nth
derivative on [0,1]. Then W is a subspace of C[0,1].
• Let W be the set of polynomials of degree at most n − 1
which are equal to 0 at x = 1. Then W is a subspace of Pn.
• Let W be the set of functions which are constant on [a, b].
Then W is a subspace of the space of bounded functions on
[a, b].
The Nullspace
Definition 2 Let A be an m×n. Then the nullspace of A, N(A)
is the subspace of Rn containing all x ∈ Rn such that Ax = 0.
The Span and the spanning set
Definition 3 Let v1, . . . ,vn ∈ V .
• Vector of the form∑n
i=1 αivi is called a linear combination
of v1, . . . ,vn.
• The span of v1, . . . ,vn, Span(v1, . . . ,vn), is the set of all linear
combinations of v1, . . . ,vn.
Theorem 1 Let v1, . . . , vn ∈ V . Then Span(v1, . . . ,vn) is a sub-
space of V .
Span(x, y)
Definition 4 Let v1, . . . ,vn ∈ V . Then the set {v1, . . . ,vn} is
called a spanning set for V if Span(v1, . . . ,vn) = V .
Remarks:
• If Span(v1, . . . ,vn) = V and v1 is a linear combination of
v2, . . . ,vn then Span(v2, . . . ,vn) = V .
• Given v1, . . . ,vn it is possible to write one of the vectors as a
linear combination of others if and only if there exist scalars
c1, . . . , cn such that
n∑
i=1
civi = 0.
Linear Independence
Definition 5 Vectors v1, . . . ,vn ∈ V are said to be linearly inde-
pendent if
n∑
i=1
civi = 0
implies that all ci = 0.
Otherwise vectors are said to be linearly dependent.
Example 3 • Show that (3,1,1)T , (1,1,0)T , (1,0,0)T are lin-
early independent in R3.
• Show that (1,1,1)T , (2,0,−2)T , (0,1,2)T are not linearly in-
dependent in R3.
• Show that f1, f2 where f1(x) = 2x + 7, f2(x) = x3 − 1 are
linearly independent in C[0,1].
• Show that 1+√
2 and 0.5 are linearly independent in Q(√
2).
• Show that ex, e−x are linearly independent in C[0,1].
Theorem 2 Let v1, . . . ,vn ∈ Rn. Then v1, . . . ,vn are linearly
independent if and only if the matrix X = (v1, . . . ,vn) is nonsin-
gular.
Theorem 3 Let v1, . . . ,vn ∈ V . Then v ∈ Span(v1, . . . ,vn) can
be written uniquely as a linear combination of v1, . . . ,vn if and
only if v1, . . . ,vn are linearly independent.
Vector Spaces of Functions
Definition 6 Let f1, . . . , fn ∈ C(n−1)[a, b]. Then W [f1, . . . , fn](x) =
det([f(i−1)j (x)]) where i, j = 1, . . . , n is called the Wronskian of
f1, . . . , fn.
Theorem 4 If f1, . . . , fn ∈ C(n−1)[a, b] are linearly dependent then
W [f1, . . . , fn](x) is the zero function.
Example 4 Let f1(x) = sin x, f2(x) = cosx and f3(x) = 2ex.
Show that f1, f2, f3 are linearly independent.
Basis and Dimension
Definition 7 The set of vectors B ⊆ V is called a basis for
V if span(B) = V and any finite subset of B contains linearly
independent vectors.
If B is finite then the above becomes: B = {v1, . . . ,vn} and
• v1, . . . ,vn are linearly independent.
• v1, . . . ,vn span V .
The standard basis for Rn
{ei|i = 1, . . . n}where
ei = (0, . . . ,0,1,0, . . . ,0).
Theorem 5 If v1, . . . ,vn span V then any set of m > n vectors
in V is linearly dependent.
Theorem 6 If v1, . . . ,vn and u1, . . . ,um are two bases for V then
m = n.
Definition 8 • If V has a finite basis then the dimension of V
is the size of basis of V and V is called finite dimensional.
• If V = {0} then dimension of V is 0.
• Otherwise V is said to be infinite-dimensional.
Theorem 7 Let V be a vector space of dimension n > 0.
• Any set of n linearly independent vectors spans V .
• Any n vectors that span V are linearly independent.
Theorem 8 Let V be a vector space of dimension n > 0.
• Any set of less than n linearly independent vectors can be
extended to a basis of V .
• Any spanning set contains a subset of n vectors that form a
basis of V .
Change of basis
Let E = [w1, . . . ,wn] and F = [v1, . . . ,vn] be two bases of V .
Problem: Given v =∑n
i=1 xiwi. Find yi so that v =∑n
i=1 yivi.
There exist sij’s such that
wi =n∑
j=1
sjivj.
and so
[w1, . . . ,wn] = [v1, . . . ,vn]S
where S = [sij] is called the transition matrix. Therefore,
S = V−1W.
If
v =n∑
i=1
xiwi
then
v =n∑
i=1
yivi
where
yi =n∑
j=1
sjixj
y = Sx.
Example 5 • Let u1 = (1,1,1)T , u2 = (1,2,2)T ,u3 = (2,3,4)T .
Find the transition matrix from [e1, e2, e3] to [u1,u2,u3].
• Find the change of basis matrix from [1, x, x2] to [1,2x,2x2−1].
Row space and Column space
Definition 9 Let A be an m × n matrix. The subspace of R1×n
spanned by rows of A is called the row space of A. The subspace
of Rm spanned by columns of A is called the column space of A.
Theorem 9 If A is row equivalent to B then the row space of
A and B are the same.
Definition 10 The rank of A is the dimension of the row space
of A.
Theorem 10 A linear system Ax = b is consistent if and only if
b is in the column space of A.
Theorem 11 An n×n matrix A is nonsingular if and only if the
column vectors of A form a basis of Rn.
Definition 11 Let A be an m × n matrix. The nullity of A is
dimension of the nullspace, N(A).
Theorem 12 (The Rank-Nullity Theorem) Let A be an m×n
matrix. Then
rank(A) + nullity(A) = n.
Theorem 13 If A is m × n then the dimension of the column
space of A and the dimension of the row space of A are the
same.