Linear Independence of Subsets: Necessary and Sufficient Conditions

[radou]

Let V be a vector space over a field F, v_{1}, \cdots, v_{n} \in V and \alpha_{1}, \cdots, \alpha_{n} \in F. Further on, let the set \left\{v_{1}, \cdots, v_{n}\right\} be linearly independent, and b be a vector defined with b=\sum_{i=1}^n \alpha_{i}v_{i}. One has to find necessary and sufficient conditions on the scalars \alpha_{1}, \cdots, \alpha_{n} such that the set S=\left\{b, v_{2}, \cdots, v_{n}\right\} is linearly independent, too.

Well, I just used a simple proposition which states that a set is dependent if there exists at least one vector from that set which can be shown as a linear combination of the rest of the vectors from the same set. So, obviously, for \alpha_{1} = 0, the set S is dependent, which makes \alpha_{1} \neq 0 a necessary condition for S to be independent. Further on, a sufficient condition would be \alpha_{1} = \cdots = \alpha_{n} = 0, which leaves us with the set S\{b}. This set must be linearly independent, since it is a subset of {v1, ..., vn}, which we know is linearly independent.

I may be boring, but I'm just checking if my reasoning is allright..

Edit. I just realized, for \alpha_{1} = \cdots = \alpha_{n} = 0 we have b = 0, which makes the set dependent! So \alpha_{1} \neq 0 is a necessary condition, but what's the sufficient condition? Is it that at least one of the scalars \alpha_{2}, \cdots, \alpha_{n}must not be equal zero?

 

[Office_Shredder]

The sufficient condition certainly includes this \alpha_{1} \neq 0

Basically, if a condition is necessary and sufficient, it means that the statement is true iff the condition is true. So search for a set of conditions that are true iff the set b, v 2-n is linearly independent.

Note that if all the alphas except \alpha_1 are zero, then b is just a multiple of v₁, and the new set is certainly linearly independent

 

[radou]

Yes, I realized that.. So, it seems \alpha_{1} \neq 0 is both necessary and sufficient condition, since the rest of the scalars can be any elements of F, all zero, all non zero, or combined.