Math 304–504
Linear Algebra
Lecture 11:
Vector spaces and their subspaces.
Vector space
Vector space is a set V equipped with two
operations α : V × V → V and µ : R × V → V
that have certain properties (listed below).
The operation α is called addition. For any
u, v ∈ V , the element α(u, v) is denoted u + v.
The operation µ is called scalar multiplication. For
any r ∈ R and u ∈ V , the element µ(r , u) is
denoted r u.
Properties of addition and scalar multiplication
A1. a + b = b + a for all a, b ∈ V .
A2. (a + b) + c = a + (b + c) for all a, b, c ∈ V .
A3. There exists an element of V , called the zero
vector and denoted 0, such that a + 0 = 0 + a = a
for all a ∈ V .
A4. For any a ∈ V there exists an element of V ,
denoted −a, such that a + (−a) = (−a) + a = 0.
A5. r (a + b) = r a + r b for all r ∈ R and a, b ∈ V .
A6. (r + s)a = r a + sa for all r , s ∈ R and a ∈ V .
A7. (rs)a = r (sa) for all r , s ∈ R and a ∈ V .
A8. 1a = a for all a ∈ V .
• Associativity of addition implies that a multiple
sum u1 + u2 + · · · + uk is well defined for any
u1 , u2 , . . . , uk ∈ V .
• Subtraction in V is defined as usual:
a − b = a + (−b).
• Addition and scalar multiplication are called
linear operations.
Given u1 , u2 , . . . , uk ∈ V and r1 , r2 , . . . , rk ∈ R,
r1 u1 + r2 u2 + · · · + rk uk
is called a linear combination of u1 , u2 , . . . , uk .
Additional properties of vector spaces
• The zero vector is unique.
• For any a ∈ V , the negative −a is unique.
• a + b = c ⇐⇒ a = c − b for all a, b, c ∈ V .
• a + c = b + c ⇐⇒ a = b for all a, b, c ∈ V .
• 0a = 0 for any a ∈ V .
• (−1)a = −a for any a ∈ V .
Examples of vector spaces
In most examples, addition and scalar multiplication
are natural operations so that properties A1–A8 are
easy to verify.
• R: real numbers
• Rn (n ≥ 1): coordinate vectors
• C: complex numbers
• Mm,n (R): m×n matrices with real entries
(also denoted Rm×n )
• R∞ : infinite sequences (x1 , x2 , . . . ), xi ∈ R
• {0}: the trivial vector space
Functional vector spaces
• P: polynomials p(x) = a0 + a1 x + · · · + an x n
en : polynomials of degree n (not a vector space)
• P
• Pn : polynomials of degree at most n
• F (R): all functions f : R → R
• C (R): all continuous functions f : R → R
• F (R) \ C (R): all discontinuous functions
f : R → R (not a vector space)
• C 1 [a, b]: all continuously differentiable functions
f : [a, b] → R
• C ∞ [a, b]: all smooth functions f : [a, b] → R
Counterexample: dumb scaling
Consider the set V = Rn with the standard
addition and a nonstandard scalar multiplication:
r ⊙ a = 0 for any a ∈ Rn and r ∈ R.
Properties A1–A4 hold because they do not involve
scalar multiplication.
A5. r ⊙ (a + b) = r ⊙ a + r ⊙ b ⇐⇒ 0 = 0 + 0
A6. (r + s) ⊙ a = r ⊙ a + s ⊙ a ⇐⇒ 0 = 0 + 0
A7. (rs) ⊙ a = r ⊙ (s ⊙ a)
⇐⇒ 0 = 0
A8. 1 ⊙ a = a
⇐⇒ 0 = a
A8 is the only property that fails. As a consequence,
property A8 does not follow from properties A1–A7.
Counterexample: lazy scaling
Consider the set V = Rn with the standard
addition and a nonstandard scalar multiplication:
r ⊙ a = a for any a ∈ Rn and r ∈ R.
Properties A1–A4 hold because they do not involve
scalar multiplication.
A5. r ⊙ (a + b) = r ⊙ a + r ⊙ b ⇐⇒ a + b = a + b
A6. (r + s) ⊙ a = r ⊙ a + s ⊙ a ⇐⇒ a = a + a
A7. (rs) ⊙ a = r ⊙ (s ⊙ a)
⇐⇒ a = a
A8. 1 ⊙ a = a
⇐⇒ a = a
The only property that fails is A6.
Subspaces of vector spaces
Definition. A vector space V0 is a subspace of a
vector space V if V0 ⊂ V and the linear operations
on V0 agree with the linear operations on V .
Examples.
• P: polynomials p(x) = a0 + a1 x + · · · + an x n
• Pn : polynomials of degree at most n
Pn is a subspace of P.
• F (R): all functions f : R → R
• C (R): all continuous functions f : R → R
C (R) is a subspace of F (R).
If S is a subset of a vector space V then S inherits
from V addition and scalar multiplication. However
S need not be closed under these operations.
Proposition A subset S of a vector space V is a
subspace of V if and only if S is nonempty and
closed under linear operations, i.e.,
x, y ∈ S =⇒ x + y ∈ S,
x ∈ S =⇒ r x ∈ S for all r ∈ R.
Proof: “only if” is obvious.
“if”: properties like associative, commutative, or distributive
law hold for S because they hold for V . We only need to
verify properties A3 and A4. Take any x ∈ S (note that S is
nonempty). Then 0 = 0x ∈ S. Also, −x = (−1)x ∈ S.
System of linear equations:

a11 x1 + a12 x2 + · · · + a1n xn = b1



a21 x1 + a22 x2 + · · · + a2n xn = b2
·········



am1 x1 + am2 x2 + · · · + amn xn = bm
Any solution (x1 , x2 , . . . , xn ) is an element of Rn .
Theorem The solution set of the system is a
subspace of Rn if and only if all equations in the
system are homogeneous (all bi = 0).
Theorem The solution set of the system is a
subspace of Rn if and only if all equations in the
system are homogeneous (all bi = 0).
Proof: “only if”: the zero vector 0 = (0, 0, . . . , 0) is a
solution only if all equations are homogeneous.
“if”: if all equations are homogeneous then the solution set is
not empty because it contains 0.
Suppose x = (x1 , x2 , . . . , xn ) and y = (y1 , y2 , . . . , yn ) are
solutions. That is, for every 1 ≤ i ≤ m
ai1 x1 + ai2 x2 + · · · + ain xn = 0,
ai1 y1 + ai2 y2 + · · · + ain yn = 0.
Then ai1 (x1 + y1 ) + ai2 (x2 + y2 ) + · · · + ain (xn + yn ) = 0
and ai1 (rx1 ) + ai2 (rx2 ) + · · · + ain (rxn ) = 0 for all r ∈ R.
Hence x + y and r x are also solutions.
Let V be a vector space and v1 , v2 , . . . , vn ∈ V .
Consider the set L of all linear combinations
r1 v1 + r2 v2 + · · · + rn vn , where r1 , r2 , . . . , rn ∈ R.
Theorem L is a subspace of V .
Proof: First of all, L is not empty. For example,
0 = 0v1 + 0v2 + · · · + 0vn belongs to L.
The set L is closed under addition since
(r1 v1 +r2 v2 + · · · +rn vn ) + (s1 v1 +s2 v2 + · · · +sn vn ) =
= (r1 +s1 )v1 + (r2 +s2 )v2 + · · · + (rn +sn )vn .
The set L is closed under scalar multiplication since
t(r1 v1 +r2 v2 + · · · +rn vn ) = (tr1 )v1 +(tr2 )v2 + · · · +(trn )vn .
Example. V = R3 .
• The plane z = 0 is a subspace of R3 .
• The plane z = 1 is not a subspace of R3 .
• The line t(1, 1, 0), t ∈ R is a subspace of R3
and a subspace of the plane z = 0.
• The line (1, 1, 1) + t(1, −1, 0), t ∈ R is not a
subspace of R3 as it lies in the plane x + y + z = 3,
which does not contain 0.
• The plane t1 (1, 0, 0) + t2 (0, 1, 1), t1 , t2 ∈ R is
a subspace of R3 .
• In general, a line or a plane in R3 is a subspace
if and only if it passes through the origin.