Week 2

Math 4377 and 6308

1.3: Subspaces

Definition: A subset 𝑊 of a vector space 𝑉 over a field 𝐹 is

called a subspace of 𝑉 if and only if 𝑊 is a vector space over 𝐹

with the operations of addition and scalar multiplication defined

on 𝑉.

Special Subspaces of a vector space 𝑽 over 𝑭: 𝑉 and {0⃗ }

(the so-call zero subspace of 𝑉, or the trivial subspace of 𝑉).

Remark: It’s not as big a pain as you might think to prove that a

subset of a vector space 𝑉 over a field 𝐹 is a subspace of 𝑉.

Why?

Several vector space properties are automatically satisfied

because 𝑊 is contained in 𝑉. More precisely, (VS1),

(VS2), (VS5), (VS6), (VS7) and (VS8) will automatically

hold because 𝑊 is contained in a vector space.

We only have to prove

1. closure under vector addition,

2. closure under scalar multiplication,

3. 𝑊 contains a zero vector (we can do this by showing

the zero vector in 𝑉 is in 𝑊),

4. each vector in 𝑊 has an additive inverse in 𝑊.

It turns out that (4) follows automatically from (2) since the

additive inverse of any vector 𝑤 in 𝑉 is (−1)𝑤.

Theorem 1.3: Let 𝑉 be a vector space over a field 𝐹, and

suppose 𝑊 is a subset of 𝑉. Then 𝑊 is a subspace of 𝑉 if and

only if

The zero vector in 𝑉 is in 𝑊.

𝑊 is closed under vector addition inherited from 𝑉. i.e. if

𝑥, 𝑦 ∈ 𝑊, then 𝑥 + 𝑦 ∈ 𝑊.

𝑊 is closed under scalar addition inherited from 𝑉. i.e. if

𝑥 ∈ 𝑊, and 𝑎 ∈ 𝐹 then 𝑎𝑥 ∈ 𝑊.

Remark: The theorem above not only provides a simple

mechanism for proving that a subset of a vector space is a

subspace, it also gives a simple mechanism for proving that a set

is a vector space, provided the set is contained in another vector

space.

Subspaces of 𝑹𝟐:

Subspaces of 𝑹𝟑:

Note: MANY vector spaces are given in Section 1.2. We will

take advantage of this information to prove that other sets are

vector spaces (by simply showing that they are subspaces of

known vector spaces).

The book gives many examples using a general field 𝑭. I will

primarily focus on 𝑹, although everything will still be true

with an arbitrary field 𝑭.

Example 1: 𝐶(𝑅)

Example 2: 𝐶1(𝑅)

Example 3: The set of real invertible matrices IS NOT a

subspace of 𝑀2×2(𝑅).

Example 4: If 𝑛 is a natural number, then the set of real

matrices with trace 0 is a subspace of 𝑀𝑛×𝑛(𝑅).

Example 5: The set 𝑈 = {𝑓 ∈ 𝐶(𝑅)| 𝑓(0) = 0} is a subspace of

𝐶(𝑅).

Example 6: (this is also Theorem 1.4 in the text) If 𝑉 is a vector

space over a field 𝐹, and 𝑈 and 𝑊 are subspaces of 𝑉, then

𝑈 ∩ 𝑊 is a subspace of 𝑉.

Example 7: If 𝑉 is a vector space over a field 𝐹, and 𝑈 and 𝑊

are subspaces of 𝑉, then 𝑈 ∪ 𝑊 is a subspace of 𝑉 if and only if

either 𝑈 ⊆ 𝑊 or 𝑊 ⊆ 𝑈.

Section 1.4: Linear Combination and Systems of Linear

Equations

Definition: Let 𝑉 be a vector space over a field 𝐹, and suppose

𝑆 is a nonempty subset of 𝑉. A vector 𝑣 ∈ 𝑉 is said to be a linear

combination of vectors in 𝑆 if and only if there are finitely many

vectors 𝑢1, … , 𝑢𝑘 ∈ 𝑆 and scalars 𝑎1, … , 𝑎𝑘 ∈ 𝐹 so that

𝑣 = 𝑎1𝑢1 + ⋯+ 𝑎1𝑢𝑘

In this case, the vector 𝑣 is also said to be a linear combination

of the vectors 𝑢1, … , 𝑢𝑘, and we call the scalars 𝑎1, … , 𝑎𝑘 the

coefficients (or weights) associated with the linear combination.

Remark: Suppose 𝑉 is a vector space over a field 𝐹, and

suppose 𝑆 is a nonempty subset of 𝑉. Then every vector in 𝑆 is a

linear combination of vectors in 𝑆, and the zero vector 0⃗ in 𝑉 is

a linear combination of vectors in 𝑆.

Example 1: Determine whether (2, −1,3) is a linear

combination of {(−2,1,4), (4, −2, −1)}.

Example 2: Find (if possible) a value of ℎ so that 2𝑥2 + ℎ𝑥 − 1

is a linear combination of 𝑥2 + 4𝑥 + 2 and −3𝑥2 + 2𝑥 − 4.

Example 3: Determine whether every element in 𝑀2×2(𝑅) can

be written as a linear combination of vectors in the set

𝑆 = {(1 00 0

) , (1 00 1

) , (0 −11 0

) , (0 10 1

)}

Definition: Suppose 𝑉 is a vector space over a field 𝐹, and

suppose 𝑆 is a nonempty subset of 𝑉. The span of 𝑆, denoted

span(𝑆), is the set of all linear combinations of the vectors in 𝑆.

For convenience, we define the span of the empty set to be {0⃗ }.

Example 4: Write the set {(𝑎 − 𝑏𝑎 + 𝑏

2𝑎 + 3𝑏) | 𝑎, 𝑏 ∈ 𝑅} as the span of

2 vectors in 𝑅3.

Example 5: Describe span((1,2)) in 𝑅2.

Theorem 1.5: Suppose 𝑉 is a vector space over a field 𝐹, and

suppose 𝑆 is a subset of 𝑉. Then span(𝑆) is a subspace of 𝑉.

Note: This is great tool for proving that a set is a vector space.

Example 6: Show that 𝑈 = {𝑎𝑥3 − 𝑏𝑥 | 𝑎, 𝑏 ∈ 𝑅} is a subspace

of 𝑃(𝑅) by showing that 𝑈 can we written as a span of vectors

in 𝑃(𝑅).

Example 7: Show that the set {(𝑎 − 𝑏𝑎 + 𝑏

2𝑎 + 3𝑏) | 𝑎, 𝑏 ∈ 𝑅} is a

subspace of 𝑅3.

Example 8: Show that the set of 2 × 2 real matrices with trace 0

is a subspace of 𝑀2×2(𝑅), by showing that it is the span of a set

of matrices in 𝑀2×2(𝑅).

Definition: A subset 𝑆 of a vector space 𝑉 generates (or spans)

𝑉 if and only if 𝑉 = span(𝑆). In this case, we also say that the

vectors of 𝑆 generate (or span) 𝑉.

Example 9: Show that the set {(1,0), (0,1)} spans 𝑅2.

Example 10: Determine whether the set

𝑌 = {(1 00 1

) , (2 00 1

) , (0 11 0

)}

spans the subspace of 𝑀2×2(𝑅) consisting of all symmetric

matrices.

Remark: The text discusses the idea of solving a system of

linear equations in the context of determining whether a vector

is a linear combination of a set of vectors. I will assume you are

capable of solving systems of linear equations. Please review

this material and see the discussion in section 1.4.