A vector space is a set V of objects, called vectors, which you can add and multiplyby scalars in a way that obeys the “usual rules of arithmetic”. Additionally, thefollowing axioms must be satisfied for all u,v,w vectors in V and scalars c and d:

1) The sum of u and v, denoted u+ v, is in V .We have closure under vector addition.

2) The scalar multiple of u by c, denoted cu is in V .We have closure under scalar multiplication.

3) There is a zero vector 0 in V such that u+ 0 = u.An additive identity exists.

4) For each u in V there is a vector �u in V such that u+ (�u) = 0.Additive inverses exist.

5) u+ v = v + u, (u+ v) +w = u+ (v +w).These are the additive axioms.

6) c(u+ v) = cu+ cv, (c+ d)u = cu+ du, c(du) = (cd)u, 1u = u.These are the distributive axioms

The space V is a collection of objects that are defined by some characteristic, andour goal is to see if we can: say for a fact that all of the axioms are satisfied, or givea counterexample showing that one or more of the axioms is not satisfied.

Example 0.1. Let V be the collection of all vectors

x

y

�such that xy � 0.

In set notation, this is written as: V =

⇢x

y

�: xy � 0

�. Is V a vector space?

1

4:1: Vector Spaces and Subspaces 2

Some examples of vector spaces are:

• vectors in Rn,

• 2⇥ 2 matrices with real-valued entries,

• all quadratic polynomials of one variable t, denoted P2.

4:1: Vector Spaces and Subspaces 3

A subspace H of a vector space V is a collection of objects such that:

1) The zero vector of V is in H.

2) H is closed under vector addition.

3) H is closed under scalar multiplication.

If these three properties are satisfied, then the subspaceH of V is itself a vector space.This says: every subspace is a vector space, and every vector space is a subspace (ofpossibly larger spaces!).

4:1: Vector Spaces and Subspaces 4

Example 0.2. Consider the vector space R3; that is, all possible vectors with 3 en-

tries,

R3 =

8<

:

2

4x1

x2

x3

3

5 : x1, x2, x3 are real numbers

9=

; .

(a) Consider all vectors in R3with x1 = x2 = x3, and let them be in a space H:

H =

8<

:

2

4x1

x2

x3

3

5 : x1 = x2 = x3

9=

; .

Is H a subspace of R3?

(b) Consider all vectors in R3with x1 = 1, and let them be in a space K:

K =

8<

:

2

4x1

x2

x3

3

5 : x1 = 1

9=

; .

Is K a subspace of R3?

4:1: Vector Spaces and Subspaces 5

(c) Consider all vectors in R3with x2 = 0, and let them be in a space H:

H =

8<

:

2

4x1

x2

x3

3

5 : x2 = 0

9=

; .

Is H a subspace of R3?

(d) Consider all vectors in R3with x1 + x2 + x3 < 1, and let them be in a space K:

K =

8<

:

2

4x1

x2

x3

3

5 : x1 + x2 + x3 < 1

9=

; .

Is K a subspace of R3?

4:1: Vector Spaces and Subspaces 6

Example 0.3. Consider the vector space M2⇥2; that is, all possible 2⇥ 2 matrices,

M2⇥2 =

⇢a b

c d

�: a, b, c, d are real numbers

�.

(e) Consider all 2⇥2 matrices in M2⇥2 with the first column as all zeros and let them

be in a space H:

H =

⇢a b

c d

�: a = 0, c = 0

�.

Is H a subspace of M2⇥2?

(f) Consider all 2 ⇥ 2 matrices in M2⇥2 with their determinant equal to 0 and let

them be in a space K:

K =

⇢a b

c d

�: ad� bc = 0

�.

Is K a subspace of M2⇥2?

4:1: Vector Spaces and Subspaces 7

Example 0.4. Given v1 and v2 in a vector space V , let H = span{v1,v2}. That is,the vectors in H are all the linear combinations of v1,v2. Show that H is a subspace

of V .

Is 0 in H?

Is H closed under vector addition?

Is H closed under scalar multiplication?

If the vectors v1,v2, . . . ,vn are in a vector space Rn, then the span of these vectorsis always a subspace of Rn.