Subspace

Subspace

Definition

A subset of is called a subspace of vector space if for every and we have:

Property Explanation
is closed under addition
addition is commutative
addition is associative
For any , there exists a vector such that zero vector
For all . there exists a such that additive inverse
is closed under scalar multiplication
scalar multiplication is associative
distributive
distributive law
scalar multiplicative identity
Intuition

A subspace is a space that is wholly contained in another. That is, each element of a space are all in another space, a similar principle to the subset. For example, a line in is always a subspace, and a point on a line is a subspace if it is on the line. Otherwise, it is not a subspace.

Subspace Test

Since is a subset of , we don't need to prove 2, 3, 7, 8, 9, and 10. We only need to verify 1, 4, 5, and 6.

Theorem

A subset is a subspace of if:

  1. contains the zero vector of :
  2. is closed under vector addition: If , then
  3. is closed under scalar multiplication: If and , then
Example

The set

is not a subspace of since

Example

Show that

is a subspace of
(See Set Builder Notation)

  1. Since ,
  2. Assume and are vectors in .
    Then , and .
    We must show that by showing that .so , and is closed under vector addition
  3. For , we must show that by showing that so , and is closed under scalar multiplication.

Thus, is a subspace of by the Subspace test.

Example

Show that

is a subspace of

  1. Taking gives
  2. Let . Then there exists such that:Which givesSo for any
  3. We haveSo .

Thus, by the Subspace test, is a subspace of .

Span is a Subspace

Theorem

Let . Then is a subspace of

(See span)

Remark

This theorem shows that we can always generate a subspace by taking the span of a finite set of vectors. In fact, every subspace of can be expressed as for some , where is a positive integer.