The question asks: What can be said about a system of vectors if it is known that the sum of all the vectors equals the zero vector?

Correct answer: The system of vectors is linearly dependent.

(Linearly dependent vectors are those for which a non-trivial linear combination gives the zero vector, meaning there exists a combination of these vectors with non-zero coefficients that sums to zero.)

Let’s break down the reasoning:

1. Linear dependence:

   • A set of vectors is linearly dependent if at least one vector in the set can be expressed as a linear combination of the others.
   • If the sum of all vectors results in the zero vector, then this indicates the presence of a linear dependence — specifically, the vectors satisfy a linear relation where not all coefficients are zero.

2. Explanation of other options:

   • Option about having a zero vector in the system:
     This is not guaranteed by the given condition. The system can be linearly dependent without containing the zero vector explicitly.
   • Options about linear independence:
     The statement about the sum being zero contradicts linear independence, which requires that the only way to get the zero vector is by setting all coefficients to zero.
   • Any subsystem is linearly dependent:
     This is not necessarily true. Even if the whole system is dependent, some subsystems could still be linearly independent.

Conclusion:

The first option, "Система векторов линейно зависима" (The system of vectors is linearly dependent), is the correct choice.

Would you like me to further elaborate on linear dependence?
Here are five related questions to explore:

1. How do we check if a system of vectors is linearly independent?
2. What is the difference between trivial and non-trivial linear combinations?
3. Can a system with more vectors than the dimension of the space be linearly independent?
4. What role does the rank of a matrix play in determining linear dependence?
5. How does linear dependence relate to span and basis?

Tip: A set of vectors in an $n$-dimensional space can have at most $n$ linearly independent vectors.