Skip to content Skip to navigation

OpenStax-CNX

You are here: Home » Content » Subspaces

Navigation

Lenses

What is a lens?

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

This content is ...

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
  • Rice Digital Scholarship

    This module is included in aLens by: Digital Scholarship at Rice UniversityAs a part of collection: "Matrix Analysis"

    Click the "Rice Digital Scholarship" link to see all content affiliated with them.

Also in these lenses

  • Lens for Engineering

    This module is included inLens: Lens for Engineering
    By: Sidney Burrus

    Click the "Lens for Engineering" link to see all content selected in this lens.

Recently Viewed

This feature requires Javascript to be enabled.
 

Subspaces

Module by: Doug Daniels, Steven J. Cox. E-mail the authors

Summary: This module defines subspaces, span, linear independence, bases and dimension.

Subspace

A subspace is a subset of a vector space that is itself a vector space. The simplest example is a line through the origin in the plane. For the line is definitely a subset and if we add any two vectors on the line we remain on the line and if we multiply any vector on the line by a scalar we remain on the line. The same could be said for a line or plane through the origin in 3 space. As we shall be travelling in spaces with many many dimensions it pays to have a general definition.

Definition 1: A subset SS of a vector space VV is a subspace of VV when
1. if xx and yy belong to SS then so does x+y x y .
2. if xx belongs to SS and tt is real then tx t x belong to SS.
As these are oftentimes unwieldy objects it pays to look for a handful of vectors from which the entire subset may be generated. For example, the set of xx for which x 1 + x 2 + x 3 + x 4 =0 x 1 x 2 x 3 x 4 0 constitutes a subspace of 4 4 . Can you 'see' this set? Do you 'see' that -1-100 -1-100 and -1010 -1010 and -1001 -1001 not only belong to a set but in fact generate all possible elements? More precisely, we say that these vectors span the subspace of all possible solutions.
Definition 2: Span
A finite collection s 1 s 2 s n s 1 s 2 s n of vectors in the subspace SS is said to span SS if each element of SS can be written as a linear combination of these vectors. That is, if for each sS s S there exist nn reals x 1 x 2 x n x 1 x 2 x n such that s= x 1 s 1 + x 2 s 2 ++ x n s n s x 1 s 1 x 2 s 2 x n s n .
When attempting to generate a subspace as the span of a handful of vectors it is natural to ask what is the fewest number possible. The notion of linear independence helps us clarify this issue.
Definition 3: Linear Independence
A finite collection s 1 s 2 s n s 1 s 2 s n of vectors is said to be linearly independent when the only reals, x 1 x 2 x n x 1 x 2 x n for which x 1 + x 2 ++ x n =0 x 1 x 2 x n 0 are x 1 = x 2 == x n =0 x 1 x 2 x n 0 . In other words, when the null space of the matrix whose columns are s 1 s 2 s n s 1 s 2 s n contains only the zero vector.
Combining these definitions, we arrive at the precise notion of a 'generating set.'
Definition 4: Basis
Any linearly independent spanning set of a subspace SS is called a basis of SS.
Though a subspace may have many bases they all have one thing in common:
Definition 5: Dimension
The dimension of a subspace is the number of elements in its basis.

Content actions

Download module as:

Add module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks