Skip to content Skip to navigation

OpenStax-CNX

You are here: Home » Content » Linear Algebra: The Basics

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.
  • OrangeGrove display tagshide tags

    This module is included inLens: Florida Orange Grove Textbooks
    By: Florida Orange GroveAs a part of collection: "Signals and Systems"

    Click the "OrangeGrove" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

  • Rice Digital Scholarship display tagshide tags

    This module is included in aLens by: Digital Scholarship at Rice UniversityAs a part of collection: "Signals and Systems"

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

    Click the tag icon tag icon to display tags associated with this content.

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.

  • richb's DSP display tagshide tags

    This module is included inLens: richb's DSP resources
    By: Richard BaraniukAs a part of collection: "Signals and Systems"

    Comments:

    "My introduction to signal processing course at Rice University."

    Click the "richb's DSP" link to see all content selected in this lens.

    Click the tag icon tag icon to display tags associated with this content.

  • Busbee's Math Materials display tagshide tags

    This module is included inLens: Busbee's Math Materials Lens
    By: Kenneth Leroy Busbee

    Click the "Busbee's Math Materials" link to see all content selected in this lens.

    Click the tag icon tag icon to display tags associated with this content.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.
 

Linear Algebra: The Basics

Module by: Michael Haag, Justin Romberg. E-mail the authors

Summary: This module will give a very brief tutorial on some of the basic terms and ideas of linear algebra. These will include linear independence, span, and basis.

This brief tutorial on some key terms in linear algebra is not meant to replace or be very helpful to those of you trying to gain a deep insight into linear algebra. Rather, this brief introduction to some of the terms and ideas of linear algebra is meant to provide a little background to those trying to get a better understanding or learn about eigenvectors and eigenfunctions, which play a big role in deriving a few important ideas on Signals and Systems. The goal of these concepts will be to provide a background for signal decomposition and to lead up to the derivation of the Fourier Series.

Linear Independence

A set of vectors x , x i Cn: x 1 x 2 x k x x i n x 1 x 2 x k are linearly independent if none of them can be written as a linear combination of the others.

Definition 1: Linearly Independent
For a given set of vectors, x 1 x 2 x n x 1 x 2 x n , they are linearly independent if c 1 x 1 + c 2 x 2 ++ c n x n =0 c 1 x 1 c 2 x 2 c n x n 0 only when c 1 = c 2 == c n =0 c 1 c 2 c n 0

Example

We are given the following two vectors: x 1 =32 x 1 3 2 x 2 =-6-4 x 2 -6 -4 These are not linearly independent as proven by the following statement, which, by inspection, can be seen to not adhere to the definition of linear independence stated above. ( x 2 =-2 x 1 )(2 x 1 + x 2 =0) x 2 -2 x 1 2 x 1 x 2 0 Another approach to reveal a vectors independence is by graphing the vectors. Looking at these two vectors geometrically (as in Figure 1), one can again prove that these vectors are not linearly independent.

Figure 1: Graphical representation of two vectors that are not linearly independent.
Figure 1 (vec_f1.png)

Example 1

We are given the following two vectors: x 1 =32 x 1 3 2 x 2 =12 x 2 1 2 These are linearly independent since c 1 x 1 =( c 2 x 2 ) c 1 x 1 c 2 x 2 only if c 1 = c 2 =0 c 1 c 2 0 . Based on the definition, this proof shows that these vectors are indeed linearly independent. Again, we could also graph these two vectors (see Figure 2) to check for linear independence.

Figure 2: Graphical representation of two vectors that are linearly independent.
Figure 2 (vec_f2.png)

Exercise 1

Are x 1 x 2 x 3 x 1 x 2 x 3 linearly independent? x 1 =32 x 1 3 2 x 2 =12 x 2 1 2 x 3 =-10 x 3 -1 0

Solution

By playing around with the vectors and doing a little trial and error, we will discover the following relationship: x 1 x 2 +2 x 3 =0 x 1 x 2 2 x 3 0 Thus we have found a linear combination of these three vectors that equals zero without setting the coefficients equal to zero. Therefore, these vectors are not linearly independent!

As we have seen in the two above examples, often times the independence of vectors can be easily seen through a graph. However this may not be as easy when we are given three or more vectors. Can you easily tell whether or not these vectors are independent from Figure 3. Probably not, which is why the method used in the above solution becomes important.

Figure 3: Plot of the three vectors. Can be shown that a linear combination exists among the three, and therefore they are not linear independent.
Figure 3 (vec_f3.png)

Hint:

A set of mm vectors in Cn n cannot be linearly independent if m>n m n .

Span

Definition 2: Span
The span of a set of vectors x 1 x 2 x k x 1 x 2 x k is the set of vectors that can be written as a linear combination of x 1 x 2 x k x 1 x 2 x k span x 1 x k = α , α i Cn: α 1 x 1 + α 2 x 2 ++ α k x k span x 1 x k α α i n α 1 x 1 α 2 x 2 α k x k

Example

Given the vector x 1 =32 x 1 3 2 the span of x 1 x 1 is a line.

Example

Given the vectors x 1 =32 x 1 3 2 x 2 =12 x 2 1 2 the span of these vectors is C2 2 .

Basis

Definition 3: Basis
A basis for Cn n is a set of vectors that: (1) spans Cn n and (2) is linearly independent.
Clearly, any set of nn linearly independent vectors is a basis for Cn n .

Example 2

We are given the following vector e i =00100 e i 0 0 1 0 0 where the 11 is always in the iith place and the remaining values are zero. Then the basis for Cn n is i ,i=12n: e i i i 1 2 n e i

Note:

i ,i=12n: e i i i 1 2 n e i is called the standard basis.

Example 3

h 1 =11 h 1 1 1 h 2 =1-1 h 2 1 -1 h 1 h 2 h 1 h 2 is a basis for C2 2 .

Figure 4: Plot of basis for C2 2
Figure 4 (vec_f4.png)

If b 1 b 2 b 1 b 2 is a basis for Cn n , then we can express any xCn x n as a linear combination of the b i b i 's: α , α i C:x= α 1 b 1 + α 2 b 2 ++ α n b n α α i x α 1 b 1 α 2 b 2 α n b n

Example 4

Given the following vector, x=12 x 1 2 writing x x in terms of e 1 e 2 e 1 e 2 gives us x= e 1 +2 e 2 x e 1 2 e 2

Exercise 2

Try and write x x in terms of h 1 h 2 h 1 h 2 (defined in the previous example).

Solution

x=32 h 1 +-12 h 2 x 3 2 h 1 -1 2 h 2

In the two basis examples above, x x is the same vector in both cases, but we can express it in many different ways (we give only two out of many, many possibilities). You can take this even further by extending this idea of a basis to function spaces.

Note:

As mentioned in the introduction, these concepts of linear algebra will help prepare you to understand the Fourier Series, which tells us that we can express periodic functions, ft f t , in terms of their basis functions, ei ω 0 nt ω 0 n t .

Download LabVIEW Source

Figure 5: video from Khan Academy, Basis of a Subspace - 20 min.
Khan Lecture on Basis of a Subspace

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