Prove that in any set of four vectors in space, at least one of the vectors is a
linear combination of the other three vectors.
i got no idea how to approach this question mainly because there are so many scenarios, all 4 in the same plane, 3 in same plane, 2 in same plane and the other 2 in same plane or the remaining 2 in different planes, etc etc. help please
Thx for your attempt to help me, but i still don't understand how u prove it though. I know that clearly any 1 of the 4 vectors in the space R^3 can be represented as a linear combination but what i don't get is how u form a "basis" with the independent vectors.
I quote from Basis (linear algebra - Wikipedia, the free encyclopedia):
"To prove that a set B is a basis for a finite-dimensional vector space V, it is sufficient to show that the number of elements in B equals the dimension of V, and one of the following:
- B is linearly independent, or
- span(B) = V."
Do you know what a basis is yet? Do you know that any basis in a vector space will have the same size as any other basis? If so you know {(1,0,0),(0,1,0),(0,0,1)} is a basis for . If there were a vector v that could not be written as a linear combination of these three basis vectors then you could form another set which necessarily is a subset of a basis {(1,0,0),(0,1,0),(0,0,1),v} which has 4 elements which contradicts the result that says every basis has the same size.
A basis is a spanning set that is linearly independant.