Are Linearly Dependent Vectors Always a Linear Combination?

[Dell]

given the vectors,

v₁,v₂...v_k+1.
show that if v₁,v₂...v_k+1. are dependent then v₁,v₂...v_k+1 are definitely dependent.

can i say that in a series of dependent vectors, at least one must be a linear combination of the others therefore, if v₁,v₂...v_k+1 are dependent, v₁,v₂...v_k+1 must be since it contains the vector which was a linear combination (from the larger series)

 

[MathematicalPhysicist]

What is definitely dependent?

 

[Mark44]

Dell,
I think you have written the problem incorrectly. My guess is that this is the problem:

Given the vectors, v1,v2,...,vk+1,
show that if v1,v2,...,vk+1 are linearly dependent, then v1,v2,...,vk are linearly dependent.

To answer your question, yes, in any collection of linearly dependent vectors, it must be the case that one of them is some linear combination of the rest.

 

[Unco]

Dell,
I think you have written the problem incorrectly. My guess is that this is the problem:

Given the vectors, v1,v2,...,vk+1,
show that if v1,v2,...,vk+1 are linearly dependent, then v1,v2,...,vk are linearly dependent.

To answer your question, yes, in any collection of linearly dependent vectors, it must be the case that one of them is some linear combination of the rest.

Huh? In R^2 the vectors (1,0), (0,1), (1,1) are linearly dependent, but (1,0) and (0,1) are linearly independent.