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 dependant then v₁,v₂...v_k+1 are definitely dependant.

can i say that in a series of dependant vectors, at least one must be a linear combination of the others therefore, if v₁,v₂...v_k+1 are dependant, 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 dependant?

 

[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.