# Thread: Linear independence of vectors

1. ## Linear independence of vectors

Hi everyone!

My question is:

If you have 4 vectors: u = [3, 2, -4] v = [-6, 1, 7] w = [0, -5, 2] z = [3, 7, -5]

and the sets {u, v}, {u, w}, {u, z}, {v, w}, {v, z}, and {w, z} are all linearly independent, than why can we not assume that the set {u, v, w, z} is linearly independent as well?

Because, there is a theorem that says that a set is linearly dependent if at least one of the vectors is a combination of the others, so if we try all of the combinations, and none of the vectors are a combination of the others, than why can we not assume that {u, v, w, z} is linearly independent?

Thank you so much for taking the time to look at this

2. ## Re: Linear independence of vectors

Originally Posted by Nora314
If you have 4 vectors: u = [3, 2, -4] v = [-6, 1, 7] w = [0, -5, 2] z = [3, 7, -5]
and the sets {u, v}, {u, w}, {u, z}, {v, w}, {v, z}, and {w, z} are all linearly independent, than why can we not assume that the set {u, v, w, z} is linearly independent as well?

Because, there is a theorem that says that a set is linearly dependent if at least one of the vectors is a combination of the others, so if we try all of the combinations, and none of the vectors are a combination of the others, than why can we not assume that {u, v, w, z} is linearly independent?
Any set of four vectors in 3-space will be linearly dependent.

3-space is spanned by a set of three linearly independent vectors.

3. ## Re: Linear independence of vectors

Thank you, I understand now, I forgot the fact that when you have 4 vectors with 3 entries each, than they will per definition be linearly dependent!