why is the empty set linearly independent? For context, I am reading P.R Halmos's Finite-Dimensional Vector Spaces's section on linear dependence. The book wrote a lot of explanation for why the empty set is linearly independent around the definition of linear dependence
Here's the definition provided in the text:
Definition. A finite set $\{x_i \}$ of vectors is linearly dependent if there exists a corresponding set $\{a_i \}$ of scalars, not all zero, such that $$\sum_i a_i x_i = 0 $$ If, on the other hand, $\sum_i a_i x_i = 0 $ implies that $a_i = 0 $ for each $ i $, the set $\{x_i \}$ is linearly independent
And the explanation for why the empty set is linearly independent as I've understood is as follows: Since there is no indices $ i $ at all for an empty set, you cannot assign to some of them a non-zero scalar, thus it's not linearly dependent.
But what I'm confused about is that the negation of "some scalars are non-zero" is "all scalars are zero". Then I can use the same argument to say that since there is no indices $ i $ at all for an empty set, you cannot assign to all the vectors a zero scalar, thus it's not linearly independent.
Especially when the text, for sake of intuition, tries to rephrase the definition of linear independence to "If $\sum_i a_i x_i = 0 $ then there is no index $ i $ for which $ a_i \neq 0 $". Here, equivalently, we can say "If $\sum_i a_i x_i = 0 $ then for all indices $ i $ , $ a_i = 0 $". I feel like this is just playing with words and did not address the problem

Say that a "dependence relation" for a finite set of vectors is a linear combination of elements from the set that is equal to zero.  As you say, the empty set of vectors satisfies a unique dependence relation, namely, the empty sum is equal to zero.  In this dependence relation, there are no coefficients.  Therefore, all the coefficients are equal to zero.  In other words: in every dependence relation satisfied by the empty set, all coefficients are equal to 0.  This is the definition of linear independence.
