Skip to content Skip to navigation

Connexions

You are here: Home » Content » Vector Spaces

Navigation

Recently Viewed

This feature requires Javascript to be enabled.
 

Vector Spaces

Module by: Richard Baraniuk. E-mail the author

Recall that we know how to add two signal/vectors and scale a vector/signal.

A vector space VV is a non empty subset of vectors such that given xx, y∈V y V , then (x+y)∈V x y V , and given x∈V x V , and a scalar αα, then α⁢x∈V α x V . In other words, a vector space is closed under addition and scaling.

note:

Depending on whether α∈R α or C we can distinguish real vectors spaces from complex vector spaces.

Example 1: Vector Spaces

R, R2 2 , R3 3 , ...... RN N

C, C2 2 , C3 3 , ...... CN N

a line in R2 2 through the origin

a line in R3 3 through the origin

a plane in R3 3 through the origin

...

0 0

Exercise 1

Vector spaces?

Figure 1
Figure 1 (spaces1.png)

A subspace of a vector space is a subset of the vectors that is itself a vector space.

Example 2

Vectors lying in a line through the origin in R2 2 :

Figure 2
Figure 2 (spaces2.png)

Exercise 2

Fundamental idea in DSP: split the vector space of all signals into "signal space" and "noise space". Why is this reasonable?

Show that the set of all linear combinations of MM vectors in RN N is a vector space.

If a vector space consists of all linear combinations of MM vectors, then these vectors span the vector space. In other words, the spanning vectors can generate a vector in the vector space.

Example 3

The vector 11 1 1 spans this subspace in R2 2 :

Figure 3
Figure 3 (spaces3.png)

Example 4

The canonical δδ basis, δ 0 ... δ N−1 δ 0 ... δ N 1 , span RN N .

Exercise 3

Can M<N M N vectors span RN N ?

Do M≥N M N vectors always span RN N ?

Exercise 4

Is RN N a subspace if CN N ?

Content actions

Download module as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

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