Determining Linear Independence: v1, v2,...vn, n≥4

[grapnell]

How do I determine this:

Problem:
The vectors: v1, v2, ... , vn, n >= 4 and are linearly independent.
Determine if the following vectors are also linearly independent.

a) the vectors v1 - v2, v2 + v3, v3 + v1

b) the vectors v1 - v2, 2(v2 - v3), 3(v3 - v4), ..., n(vn - v1)

c) the vectors v1, v1 + c1v2, v2+c2v3, v3 + c3v4, ..., vn +cnv1
where c1, ... cn are real numbers

If someone could give me a semi detailed to detailed explanation on ow to go about solving a problem like this, rather than just an answer, I would greatly appreciate it very much.

Thanks in advance,
Jeff

 

[Hurkyl]

Linear independence / dependence is all about what linear combinations of the vectors sum to zero, right? It would seem that starting down by writing down a general linear combination of your vectors would be a good place to start...

 

[grapnell]

What exactly do you mean by "general" linear combination of my vectors?
Thanks,
Jeff

 

[HallsofIvy]

Rule 1: Learn the definitions!

You can't possibly prove that a set of vectors is or is not "independent" without knowing the definition of "independent".

I'll bet your book defines "independent" in terms of a "linear combination" of vectors so you can't understand "independent" until you know the definition of "linear combination".

Your assignment: go to your textbook and look up the definitions of "independent" and "linear combination".

 

[grapnell]

I already know what that means, I just don't know what you mean by "GENERAL" my teacher uses really weird notation, completely different than what the book uses, plus he barely speaks english, so I am having to learn all of this on my own. If you could just show me and explain to me how to do the problem, I think I can probably pick it up, I just need to figure out where to start. I can easily figure out if a set of vectors is linearly dependent or independent if i know what values are in the vectors, but for some reason, I am just not clicking with this concept in this particular problem, and there are no problems like them in the book, or in any of the other 7 books that I rented from the library.
Thanks and I appreciate your help
Jeff

 

[Hurkyl]

For example, can you write down an expression that represents any linear combination of the two vectors U and V?


edit: Frederick beat me to it - I was using "general" in its English sense.

 

[Fredrik]

"General" just means "not specific". Hurkyl is just suggesting that you write down an expression that can represent any linear combination of the vectors in your problems.

From the definition of "linearly independent" it's trivial to see that if and only if a set of vectors are not linearly independent, you can express anyone of them as a linear combination of the others. Hence, it might be a good idea to see if any of your vectors can be expressed as a linear combination of the others. In part a, for example, what is the sum of the first and second vectors?

 

[grapnell]

It doesn't appear as though I can from the given problem. My conclusion is that none of them (a, b or c) are linearly dependent, and are all three linearly independent, because I don't see any vectors that are linear combinations of any other vectors of the set. it just seems too simple. I just feel like I am overlooking something here.

Thanks,
Jeff

 

[Hurkyl]

Sometimes it does help to invent numbers to help figure out the method. Actually, with care you can use numbers for the proof! (but you'll lose this chance to try and develop your algebraic intuition)


For the simplest case, try n = 4 and v1 = (1, 0, 0, 0)^t, v2 = (0, 1, 0, 0)^t, ...

 

[grapnell]

Thanks everyone, I just hit a break through on this... I completely understand now, and you guys are great, thanks for not just giving me the answer, but helping me figure it out. I now know what to do. I just needed to hear it in "plain english"
Thanks again
Jeff