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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 togain a deep insight into linear algebra. Rather, this brief introduction to some of the terms and ideas of linear algebra ismeant to provide a little background to those trying to get a better understanding or learn about eigenvectors andeigenfunctions, which play a big role in deriving a few important ideas on Signals and Systems. The goal of theseconcepts 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 n x 1 x 2 x k are linearly independent if none of them can be written as a linear combination of the others.

Linearly Independent
For a given set of vectors, x 1 x 2 x n , they are linearly independent if c 1 x 1 c 2 x 2 c n x n 0 only when c 1 c 2 c n 0
Graphical representation of two vectors that are not linearly independent.

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

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Graphical representation of two vectors that are linearly independent.

Are x 1 x 2 x 3 linearly independent? x 1 3 2 x 2 1 2 x 3 -1 0

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

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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 thesevectors are independent from [link] . Probably not, which is why the method used in the above solution becomesimportant.

Plot of the three vectors. Can be shown that a linear combination exists among the three, and therefore they are not linear independent.

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


The span of a set of vectors 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 span x 1 x k α α i n α 1 x 1 α 2 x 2 α k x k


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

We are given the following vector e i 0 0 1 0 0 where the 1 is always in the i th place and the remaining values are zero. Then the basis for n is i i

    1 2 n
e i
i i
    1 2 n
e i is called the standard basis .

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h 1 1 1 h 2 1 -1 h 1 h 2 is a basis for 2 .

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Plot of basis for 2

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

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

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

x 3 2 h 1 -1 2 h 2

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

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, f t , in terms of their basis functions, ω 0 n t .

Khan lecture on basis of a subspace

video from Khan Academy , Basis of a Subspace - 20 min.

Questions & Answers

how can chip be made from sand
Eke Reply
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Missy Reply
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Lale Reply
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Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
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Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
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Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
has a lot of application modern world
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Crow Reply
what about nanotechnology for water purification
RAW Reply
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yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
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industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
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What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
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Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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Source:  OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
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