
#19
Dec212, 05:51 PM

P: 5,462

When you check such a book sometimes, as in this case, you need to check more than one definition.
You will find the mention of sets in the crossreferenced definition of 'linear combination'. You have to follow the chain of definitions through. However the entry did reference sets in another way as well. 



#20
Dec212, 06:02 PM

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#21
Dec212, 06:19 PM

P: 5,462

So let us start with the definition given of linear combination.
It specifies a method of combining elements of a set. It does not prohibit those elements being sets, numbers or other elements capable of forming combinations according to the rules specified. It does allow those elements to be vectors and uses this case as an example. But it does not restrict those elements to being vectors. I am quite sure that had the authors wanted this restriction they would have specified as such, as for instance they have done in their definition of a linear mapping. Since it specifies the word elements in the plural it implies that there is more than one, which tallies with one of my earlier comments. I am equally sure that I have seen many (mostly engineering it is true) books that talk of linearly dependent equations. Along the lines, for instance of 3x+2y=6 and 12x+8y=24 are linearly dependent since there exists a coefficient (4) that you can multiply equation 1 by to obtain equation 2. The above example is very obvious but some are not and I was going to develop this to help the OP when these other issues arose. Please also look at the second half of my post#13, where I note the definition scheme is the complement of that offered by Fredrik and also give my reasons for preferring it. 



#22
Dec212, 06:30 PM

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So, for sets and equations, how do you define these notions? 



#23
Dec212, 07:22 PM

P: 51

To keep it simple, a linear combination is just the product of the columns of matrix A and some corresponding entry as weights from the column vector x. In other wards it's just Ax.




#24
Dec312, 12:38 AM

P: 783

Studiot, I'm quite sure that equations/planes/spaces cannot be added and subtracted. Only vectors and matrices can, and hence only vectors and matrices are expressible as linear combinations of one another. My professor has drilled this to my head quite thoroughly, and I find the concept quite convenient from a rigorous point of view
BiP 



#25
Dec312, 12:55 AM

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P: 16,554

To be honest, I am quite interested in the "sets being linearly independent"idea. Too bad Studiot doesn't have any references except an entry in a math encyclopedia. So if anybody has some actual references, I would be very happy to read about it!! 



#26
Dec312, 03:28 AM

P: 5,462

Going back to my post# taking equations1,2 and adding a third 3x+y=18 I can: Form the linear combinations: 4(equation1) + (1)(equation2) this is seen to equal 0 so the equations are linearly dependent and no solution is possible. 1(equation1) + (1)(equation3) this is seen to be ≠ 0 and therefore the equations can be solved for x and y Nothing in the Borowski definition implies we need the entire space of expressions of the form gx+hy=k to be able to perform these manipulations, although Fredrik's definition does seem to suggest this, though I won't deny that such a space is useful. 



#27
Dec312, 03:34 AM

P: 5,462

halo31 has put is pretty shortly and the equations I presented can be expressed in that format, but the expression 3x+2y is neither a matrix nor a vector. Of course you can add equations. The technique is extremely widely used in Physics and Engineering and of great importance. It is called superposition. Further if those equations represent some region then that is equivalent to adding those regions, but not all equations represent regions and not all regions have single equations. 



#28
Dec312, 05:48 AM

P: 422

I wish I could find this book again and see if it relates... 



#29
Dec312, 07:27 AM

P: 5,462

Kreysig p53? Griffel p89? Gupta 2.23, 1.17? Hoffman and Kunze p40? My post was written first and I used Borowski to check my statements against. 



#30
Dec312, 07:37 AM

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PF Gold
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It's fairly obvious that the intersection of two linearly independent sets is either empty or linearly independent. Maybe that's the result you had in mind. 



#31
Dec312, 07:48 AM

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PF Gold
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#32
Dec312, 07:49 AM

P: 5,462

The set of all values of p, q for which 3p+2q=6 and the set of all values for which 12p+8q=24 Put these into your x,y format and you can see that you have two sets which are linearly dependent, since they are essentially the same set. 



#33
Dec312, 08:16 AM

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PF Gold
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Do you mean that I should write these two lines as ##K=\{(p,q)\in\mathbb R^23p+2q=6\}## and ##L=\{(p,q)\in\mathbb R^212p+8q=24\}##? I wouldn't say that K and L are linearly dependent. I would just say that they're equal.




#34
Dec312, 09:41 AM

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P: 16,554

As for Nering and Hoffman & Kunze: So the notion defined here is the linear independence of a set. I do not see a definition here of the linear independence of two sets or the linear independence of equations. These definitions are perfectly compatible with what Fredrik has said. So none of these books actually agree with what you are saying. No offense, but I am starting to think that you are just misunderstanding the entire concept. 



#35
Dec312, 09:42 AM

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#36
Dec312, 09:52 AM

P: 5,462

Why are you now asking for a zero set? A vector is a set of points that satisfy certain conditions, specific to the problem in hand. Since this is getting further and further from the OP and personal to boot I withdraw from this thread. 


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