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#1 2016-07-10 09:48:36

eleanor
Member
Registered: 2016-07-10
Posts: 5

can someone explain me linear combinations and spans?

i began learning linear algebra on this site and on khan academy site and made some quizzes about vectors but when i reached the part about "Linear combinations and spans" and "Linear dependence and independence" i had a serious problem to understand it. the only thing i think i got(or hope i got it right) is that a span is all the possible vectors for a combination of two or more vectors.
and by combination its means adding them together plus any kind of multiplication on them. i am right her?. and what about "dependence/independence" stuff?.

there is an article on this site about it? i realy hope to find some easy article about it that is dedicated to people who are new to linear algebra.

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#2 2016-07-10 12:53:44

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: can someone explain me linear combinations and spans?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#3 2016-07-10 17:40:48

eleanor
Member
Registered: 2016-07-10
Posts: 5

Re: can someone explain me linear combinations and spans?

thnx for helping, can you give me some example about the relation of this to the subject and explain me shortly about the subject? i am quite lost here

Last edited by eleanor (2016-07-10 17:41:45)

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#4 2016-07-10 18:03:04

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: can someone explain me linear combinations and spans?

Let us do the independent - dependent part first.

If you have two equations like these
    x + y = 3 and 2x + 2y = 6 we can see that the second equation is just a multiple of the first equation. They are dependent. There are three types of situations that can come up.

Take a look here for an explanation.

http://hotmath.com/hotmath_help/topics/ … stems.html


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#5 2016-07-11 04:31:36

eleanor
Member
Registered: 2016-07-10
Posts: 5

Re: can someone explain me linear combinations and spans?

bobbym wrote:

Let us do the independent - dependent part first.

If you have two equations like these
    x + y = 3 and 2x + 2y = 6 we can see that the second equation is just a multiple of the first equation. They are dependent. There are three types of situations that can come up.

Take a look here for an explanation.

http://hotmath.com/hotmath_help/topics/ … stems.html

its reminds me that i saw in the video's something about vector cA(with c being a scalar) having the span of a line(or something like that).
so basicaly the span of the set of vectors is determined by if the vectors can be made by a multiplication of one vector?. if so how it is related to the three types of situations in the article?.

sorry if i am getting confused or missing something obvious.

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#6 2016-07-11 04:50:46

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: can someone explain me linear combinations and spans?

First things first, did you understand everything about this http://hotmath.com/hotmath_help/topics/ … stems.html ?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#7 2016-07-11 07:35:13

eleanor
Member
Registered: 2016-07-10
Posts: 5

Re: can someone explain me linear combinations and spans?

bobbym wrote:

First things first, did you understand everything about this http://hotmath.com/hotmath_help/topics/ … stems.html ?

i think so, i know that a system of equation have infinite solutions when one of the equation is just a multiplication of the other and have no solution when they are the same lines(same slope)
but in different positions and parallel to each other, and that as long they are different lines they going to have one solution and i think it happens as long as the x in one equation have a different coefficient from the x in the other equation.

the case with one solution is related to a case when a vector combination have an R2 span and is "independent"?

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#8 2016-07-11 10:27:26

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: can someone explain me linear combinations and spans?

Hi;

The last statement

the case with one solution is related to a case when a vector combination have an R2 span and is "independent"?

I do not know.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#9 2016-07-11 11:06:00

eleanor
Member
Registered: 2016-07-10
Posts: 5

Re: can someone explain me linear combinations and spans?

bobbym wrote:

Hi;

The last statement

the case with one solution is related to a case when a vector combination have an R2 span and is "independent"?

I do not know.

can you explain me the connection between this article and the subject of linear span's? i realy needs it cause i am going to learn linear algebra in a university next week and i don't want to get stuck with this.
anyway thank you for the help you already gave me, i understand it a bit more now but i am still confused.

Last edited by eleanor (2016-07-11 11:07:16)

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#10 2016-07-11 17:07:24

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: can someone explain me linear combinations and spans?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#11 2016-07-15 01:28:07

Bob
Administrator
Registered: 2010-06-20
Posts: 10,053

Re: can someone explain me linear combinations and spans?

hi Eleanor

I'll have a go at trying to explain.  Let's say we have a 3 dimensional space of all possible vectors.  I'll write my vectors across the line to make typing easier. The basis (1,0,0); (0,1,0);(0,0,1) is said to span the space because you can construct a linear combination from the basis to make any possible vector in the space.

eg.  (2,3,4) = 2(1,0,0) + 3(0,1,0) + 4(0,0,1)

The basis doesn't have to be that simple.  It could be (1,1,0); (0,0,1); (1,-1,0)

In that case (2,3,4) = 2.5(1,1,0) + 4(0,0,1) -0.5(1,-1,0)

check    2.5(1,1,0) + 4(0,0,1) -0.5(1,-1,0)  = (2.5,2.5,0) + (0,0,4) - (0.5,-0.5,0) = (2.5 + 0 -0.5, 2.5 + 0 +0.5, 0 + 4 + 0) = (2,3,4) as required.

To make a basis there must be exactly as many vectors as the size of the space ... so 3D needs 3 vectors and 2D needs 2 vectors.  But it is not enough just to have sufficient vectors.  None must be dependant on the others ... or to put it the other way round, the basis vectors must be independent of each other.

Let's see what happens if they are not independent.

Say I choose (1,0,0); (0,1,1); and (1,1,1)  Note (1,0,0) + (0,1,1) = (1,1,1) so the third is a linear combination of the first two.

Now let's try to make a linear combination to make (2,3,4)

a(1,0,0) + b(0,1,1) + c(1,1,1) = (2,3,4)  Can I find a, b and c ?

We have
a+c = 2
b+c = 3
b+c = 4

Clearly the second and third are inconsistent with each other.  b+c cannot be 3 and also 4.  So no solution exists.  If you tried plotting lines you would find these two equations lead to parallel lines and so would not cross.

I'll change the third vector to one that is independent of the other two.  The first two combined can make any vectors of the form (a, b, b) so I only need to choose my third vector so that its 'y' and 'z' components are different.  eg.  (1,2,3)

So I should be able to make a linear combination out of these.

a(1,0,0) + b(0,1,1) + c(1,2,3) = (2,3,4)  Can I find a, b and c ?

a+c = 2
b+2c = 3
b+ 3c = 4

Subtracting the second from the third we get c = 1, which means b = 1, and finally a = 1 from the first equation.

check 1(1,0,0) + 1(0,1,1) + 1(1,2,3) = (1+1, 1+2, 1 +3) = (2,3,4)

Hope this helps.  Post again if you need more help.

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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