|Math 231-Fall 2019|Section 4.

2|Page 1

Section 4.2: Subspaces1

Concepts:
• Subspace.
• Zero subspace.
• Examples of subspaces.
• Linear combination.
• Span.
• Solution space.

Learning Outcomes.
After completing this section, you should be able to:
• Determine whether a subset of a vector space is a subspace.
• Show that a subset of a vector space is a subspace.
• Show that a nonempty subset of a vector space is not a subspace by demonstrating
that the set is either not closed under addition or not closed under scalar
multiplication.
• Given a set 𝑆𝑆 of vectors in ℝ𝑛𝑛 and a vector 𝐯𝐯 in ℝ𝑛𝑛 , determine whether 𝐯𝐯 is a linear
combination of the vectors in 𝑆𝑆.
• Given a set 𝑆𝑆 of vectors in ℝ𝑛𝑛 , determine whether the vectors in 𝑆𝑆 span ℝ𝑛𝑛 .

1
The materials of these lecture notes are based on the textbook of the course.
 |Math 231-Fall 2019|Section 4.2|Page 2
It is often the case that some vector space is contained within a larger vector space whose
properties are known. In this section we will show how to recognize when this is the case,
we will explain how the properties of the larger vector space can be used to obtain
properties of the smaller vector space.

Subspaces
A subset 𝑊𝑊 of a vector space 𝑉𝑉 is called a subspace of 𝑉𝑉 if 𝑊𝑊 is itself a vector space under
the addition and scalar multiplication defined on 𝑉𝑉.

Remark.
In general, to show that a nonempty set 𝑊𝑊 with two operations is a vector space one
must verify the ten vector space axioms.

If 𝑊𝑊 is a subspace of a known vector space 𝑉𝑉, then to prove that it is a subspace of 𝑉𝑉, we
need to verify only the following axioms:
Axiom 1: Closure of 𝑊𝑊 under addition
Axiom 4: Existence of a zero vector in 𝑊𝑊
Axiom 5: Existence of a negative in 𝑊𝑊 for every vector in 𝑊𝑊
Axiom 6: Closure of 𝑊𝑊 under scalar multiplication

However, the next theorem shows that if Axiom 1 and Axiom 6 hold in 𝑊𝑊, then Axioms 4
and 5 hold in 𝑊𝑊 as a consequence and hence need not be verified.

Theorem.
If 𝑊𝑊 is a set of one or more vectors in a vector space 𝑉𝑉, then 𝑊𝑊 is a subspace of 𝑉𝑉 if and
only if the following conditions are satisfied.
(a) If 𝐮𝐮 and 𝐯𝐯 are vectors in 𝑊𝑊, then 𝐮𝐮 + 𝐯𝐯 is in 𝑊𝑊.
(b) If 𝑘𝑘 is a scalar and 𝐮𝐮 is a vector in 𝑊𝑊, then 𝑘𝑘𝐮𝐮 is in 𝑊𝑊.

Remark. To show that 𝑾𝑾 is not a subspace of a vector space 𝑽𝑽:

1) Check if 𝟎𝟎 ∉ 𝑊𝑊.
2) Find a vector 𝐮𝐮 ∈ 𝑊𝑊 but −𝐮𝐮 ∉ 𝑊𝑊.
Find any two vectors 𝐮𝐮, 𝐯𝐯 ∈ 𝑊𝑊 but 𝐮𝐮 + 𝐯𝐯 ∉ 𝑊𝑊.
 |Math 231-Fall 2019|Section 4.2|Page 3

Example. If 𝑉𝑉 is any vector space and if 𝑊𝑊 = {𝟎𝟎}. Show that 𝑊𝑊 is a subspace of the
vector space 𝑉𝑉.

Example. If 𝑉𝑉 = ℝ2 and if 𝑊𝑊 = {(𝑥𝑥, 𝑦𝑦) ∈ ℝ2 | 𝑥𝑥 ≥ 0, 𝑦𝑦 ≥ 0}. Show that 𝑊𝑊 is not a

subspace of the vector space 𝑉𝑉.
 |Math 231-Fall 2019|Section 4.2|Page 4

Notation.
The set 𝑉𝑉 of all 𝑚𝑚 × 𝑛𝑛 matrices with the usual matrix operations of addition and scalar
multiplication is a vector space. We will denote this vector space by the symbol 𝑀𝑀𝑚𝑚𝑚𝑚 .

Example. If 𝑉𝑉 = 𝑀𝑀𝑛𝑛𝑛𝑛 and if 𝑊𝑊 = {𝐴𝐴 ∈ 𝑀𝑀𝑛𝑛𝑛𝑛 | 𝐴𝐴𝑇𝑇 = 𝐴𝐴}. Show that 𝑊𝑊 is a subspace of
the vector space 𝑉𝑉.

Example. If 𝑉𝑉 = 𝑀𝑀𝑛𝑛𝑛𝑛 and if 𝑊𝑊 = {𝐴𝐴 ∈ 𝑀𝑀𝑛𝑛𝑛𝑛 | 𝐴𝐴−1 exists}. Determine whether 𝑊𝑊 is a

subspace of the vector space 𝑉𝑉 or not.
 |Math 231-Fall 2019|Section 4.2|Page 5

Example. If 𝑉𝑉 = 𝑀𝑀22 and if 𝑊𝑊 = {𝐴𝐴 ∈ 𝑀𝑀22 | |𝐴𝐴| = 0}. Determine whether 𝑊𝑊 is a

subspace of the vector space 𝑉𝑉 or not.

Example. If 𝑉𝑉 = 𝐹𝐹 (−∞, ∞) and if 𝑊𝑊 = 𝐶𝐶 (−∞, ∞) is set of all continuous functions on

(−∞, ∞). Determine whether 𝑊𝑊 is a subspace of the vector space 𝑉𝑉 or not.

Example. If 𝑉𝑉 = 𝐹𝐹 (−∞, ∞) and if 𝑊𝑊 = 𝐶𝐶1 (−∞, ∞) is set of all continuously

differentiable functions on (−∞, ∞). Determine whether 𝑊𝑊 is a subspace of the vector
space 𝑉𝑉 or not.
 |Math 231-Fall 2019|Section 4.2|Page 6

Notation.
For 𝑛𝑛 ≥ 0, the set 𝑉𝑉 of all polynomials of degree less than or equal 𝑛𝑛 with the usual
function operations of addition and scalar multiplication is a vector space. We will denote
this vector space by the symbol ℙ𝑛𝑛 .

Example. If 𝑉𝑉 = ℙ2 and if 𝑊𝑊 = {𝑝𝑝(𝑥𝑥) ∈ ℙ2 | degree(𝑝𝑝) = 2}. Determine whether 𝑊𝑊 is

a subspace of the vector space 𝑉𝑉 or not.
 |Math 231-Fall 2019|Section 4.2|Page 7

Example (Solution Spaces of Homogeneous Systems).

If 𝑉𝑉 = ℝ𝑛𝑛 , 𝐴𝐴 is an 𝑚𝑚 × 𝑛𝑛 matrix, and
𝑊𝑊 = {𝐱𝐱 ∈ ℝ𝑛𝑛 | 𝐴𝐴𝐱𝐱 = 𝟎𝟎}
Determine whether 𝑊𝑊 is a subspace of the vector space 𝑉𝑉 or not.

Example. If 𝑉𝑉 = ℝ𝑛𝑛 , 𝐴𝐴 is an 𝑚𝑚 × 𝑛𝑛 matrix, 𝐛𝐛 ≠ 𝟎𝟎 is an 𝑚𝑚 × 1 vector, and

𝑊𝑊 = {𝐱𝐱 ∈ ℝ𝑛𝑛 | 𝐴𝐴𝐱𝐱 = 𝐛𝐛}
Determine whether 𝑊𝑊 is a subspace of the vector space 𝑉𝑉 or not.
 |Math 231-Fall 2019|Section 4.2|Page 8

Theorem.
If 𝑊𝑊1 , 𝑊𝑊2 , … , 𝑊𝑊𝑚𝑚 are subspaces of a vector space 𝑉𝑉, then the intersection of these
subspaces
𝑊𝑊1 ⋂𝑊𝑊2 ⋂ ⋯ ⋂𝑊𝑊𝑚𝑚
is also a subspace of 𝑉𝑉.

Remark.
If 𝑊𝑊1 and 𝑊𝑊2 are subspaces of a vector space 𝑉𝑉, then the union of these subspaces
𝑊𝑊1 ⋃𝑊𝑊2
is not necessary a subspace of 𝑉𝑉.

Sometimes we will want to find the “smallest” subspace of a vector space 𝑉𝑉 that contains
all of the vectors in some set of interest. The following definition will help us to do that.

Linear Combination
If 𝐰𝐰 is a vector in a vector space 𝑉𝑉, then 𝐰𝐰 is said to be a linear combination of the vectors
𝐯𝐯1 , 𝐯𝐯2 , … , 𝐯𝐯𝑟𝑟 in 𝑉𝑉 if 𝐰𝐰 can be expressed in the form
𝐰𝐰 = 𝑘𝑘1 𝐯𝐯1 + 𝑘𝑘2 𝐯𝐯2 + ⋯ + 𝑘𝑘𝑟𝑟 𝐯𝐯𝑟𝑟
where 𝑘𝑘1 , 𝑘𝑘2 , … , 𝑘𝑘𝑟𝑟 are scalars. These scalars are called the coefficients of the linear
combination.

Example. Consider the vectors

𝐮𝐮 = (1,2, −1) and 𝐯𝐯 = (6,4,2)
3
in ℝ . Show that 𝐰𝐰 = (9,2,7) is a linear combination of 𝐮𝐮 and 𝐯𝐯.
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Example. Consider the vectors

𝐮𝐮 = (1,2, −1) and 𝐯𝐯 = (6,4,2)
3
in ℝ . Show that 𝐰𝐰 = (4, −1,8) is not a linear combination of 𝐮𝐮 and 𝐯𝐯.

Span
If 𝑆𝑆 = {𝐰𝐰1 , 𝐰𝐰2 , … , 𝐰𝐰𝑟𝑟 } is a nonempty set of vectors in a vector space 𝑉𝑉 and 𝑊𝑊 is the set of
all possible linear combinations of the vectors in 𝑆𝑆, then we say that the vectors
𝐰𝐰1 , 𝐰𝐰2 , … , 𝐰𝐰𝑟𝑟 span 𝑊𝑊. We write
𝑊𝑊 = span{𝐰𝐰1 , 𝐰𝐰2 , … , 𝐰𝐰𝑟𝑟 } or 𝑊𝑊 = span(𝑆𝑆).

Remark.
• 𝑊𝑊 = span(𝑆𝑆) is a subspace of 𝑉𝑉.
• The subspace 𝑊𝑊 is the “smallest” subspace of 𝑉𝑉 that contains all the vectors in 𝑆𝑆.
• span(𝑆𝑆) is called the subspace of 𝑉𝑉 generated by 𝑆𝑆.
• 𝐮𝐮 ∈ span{𝐰𝐰1 , 𝐰𝐰2 , … , 𝐰𝐰𝑟𝑟 } ⇔ 𝐮𝐮 = 𝑘𝑘1 𝐰𝐰1 + 𝑘𝑘2 𝐰𝐰2 + ⋯ + 𝑘𝑘𝑟𝑟 𝐰𝐰𝑟𝑟 , 𝑘𝑘1 , 𝑘𝑘2 , … , 𝑘𝑘𝑟𝑟 ∈ ℝ

The Standard Unit Vectors in ℝ𝒏𝒏 :

𝐞𝐞1 = (1,0,0, … ,0,0), 𝐞𝐞2 = (0,1,0, … ,0,0), …, 𝐞𝐞𝑛𝑛 = (0,0,0, … ,0,1) ∈ ℝ𝑛𝑛

The Standard Unit Vectors in ℝ𝟑𝟑 :

𝐢𝐢 = (1,0,0), 𝐣𝐣 = (0,1,0), 𝐤𝐤 = (0,0,1) ∈ ℝ3
 | M a t h 2 3 1 - F a l l 2 0 1 9 | S e c t i o n 4 . 2 | P a g e 10

Example. Show that span{𝐢𝐢, 𝐣𝐣, 𝐤𝐤} = ℝ3 .

Example. Show that span{1, 𝑥𝑥, 𝑥𝑥 2 , … , 𝑥𝑥 𝑛𝑛 } = ℙ𝑛𝑛 .

Example. Determine whether the vectors

𝐯𝐯1 = (1,1,2), 𝐯𝐯2 = (1,0,1), and 𝐯𝐯3 = (2,1,3)
span the vector space ℝ3 .
 | M a t h 2 3 1 - F a l l 2 0 1 9 | S e c t i o n 4 . 2 | P a g e 11

Example. Determine whether the vectors

𝑣𝑣1 = (1,1,0), 𝑣𝑣2 = (1,0,1), and 𝑣𝑣3 = (2,1,3)
span the vector space ℝ3 .
 | M a t h 2 3 1 - F a l l 2 0 1 9 | S e c t i o n 4 . 2 | P a g e 12

Example. Suppose that

𝐯𝐯1 = (1,1,0,2), 𝐯𝐯2 = (1,0,1, −1), and 𝐯𝐯3 = (1, −1,2,1).
Is the vector 𝐰𝐰 = (2,2,4,9) in span{𝐯𝐯1 , 𝐯𝐯2 , 𝐯𝐯3 }.