Linear (in)dependence

May 2017
91
0
Slovenia
Hi guys,
I'm stuck with a problem which is, I believe, simple, but I can't solve it by myself.

So, I have 4 vectors a,b,c,d that are linearly independent. I am supposed to check whether the following vectors are dependent or independent:

a1 = a+2c+d
b1 = 2a-b+c
c1 = a+c+d

Thank you in advance. :D
 
Last edited by a moderator:

SDK

Sep 2016
704
469
USA
The question makes no sense. A set of vectors can be linearly independent or not. This doesn't mean anything for single vectors. Technically, you could talk about the singleton set containing each vector but then the answer is trivial. A vector is independent if and only if it is nonzero. This is certainly not what you mean to be asking here.
 

skipjack

Forum Staff
Dec 2006
21,317
2,384
Let's assume that the set {a, b, c, d} is linearly independent, and that the problem is to check whether the set {a1, b2, c1} is linearly independent.

Note that 3c1 - a1 - b1 = b + 2d.
 
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May 2017
91
0
Slovenia
Let's assume that the set {a, b, c, d} is linearly independent, and that the problem is to check whether the set {a1, b2, c1} is linearly independent.

Note that 3c1 - a1 - b1 = b + 2d.
And that meaning?
 

SDK

Sep 2016
704
469
USA
Let's assume that the set {a, b, c, d} is linearly independent, and that the problem is to check whether the set {a1, b2, c1} is linearly independent.

Note that 3c1 - a1 - b1 = b + 2d.
Jeez I'm dumb. I don't know why I didn't realize that is what was being asked. It's pretty obvious now.