<s>[INST] Let X be a set of vectors in a vector space V over the field F.
I want to know if X is linearly dependent or independent.
So I check the linear combination expression
$$
0 = a_1\mathbf{x}_1 + a_2\mathbf{x}_2 + \cdots + a_n\mathbf{x}_n
$$
If this expression has a non-trivial solution, i.e. not all $a_i$’s are zeros, then the set X is linearly dependent.
If this expression has only the trivial solution, i.e. all $a_i$’s are zeros, then the set X is linearly independent.
Here is my question:
Suppose that there is a non-trivial solution to the expression above, that is, not all $a_i$’s are zeros.
Can we say that at least one of the vectors in X is equal to the linear combination of the other vectors in X?[/INST]