Is the zero element and zero vector the same thing in the context of Linear Independence?

Question

Say..

A finite set of elements from a linear space are linearly independent if:

1. Their trivial linear combination is equal to the zero element
2. Their trivial linear combination is equal to the zero vector

I am not sure if statements 1 and 2 say the same thing.

I know that the zero element has the properties; for elements x and y of a linear space :

x + zero element = x

x + y = zero element

And I believe a zero vector would be an n tuple with only zero elements: (0,0,...,0)

Right now I believe they are the same thing in terms of properties that they have when talking about linear spaces and I think it is just different terminology.

But I also know from an example of a field with 2 elements (even and odd) that, even is the zero element in the field and it is not an n tuple with only zero elements.

As even + even = even and even + odd = odd

Answer 1

Both your definitions are wrong. It should be, if the set is $\{v_1,\ldots,v_k\}$,

$$\sum_{i=1}^k \lambda_iv_k=0 \;\Rightarrow\;\text{all $\lambda_i=0$.}$$

You are correct that the zero vector in $\mathbb{R}^n$ has exactly $n$ entries that are equal to the real number $0$.

Finally, the terms zero element and zero vector are often used interchangeably in this context.

Answer 2

The most conscise definition of vector space is:

an abelian group on which a field acts

The zero vector of such a vector space is nothing else but the zero (or neutral) element of said abelian group. Specifically, in the vector space $F^n$ of $n$-tuples of elements of the field $F$, the zero vector is also the tuple with all entries zero, $(0,0,\ldots,0)$.

Noe however, that we always also have the ground field in the context of vector spaces. The additive group of that filed of course also has a zero element, which (unless we consider the field as vector space over itself) is not referred to as zero vector.