When are vectors considered to be linearly independent?

Vectors are considered to be linearly independent when the equation formed by their linear combination equals zero has only the trivial solution, which is when all coefficients in that combination are zero. This means that no vector in the set can be expressed as a linear combination of the others. In simpler terms, if you cannot write any vector as a mix of others, then they are independent. This property indicates that the vectors provide unique directions in the vector space they span.

The alternative options do not accurately reflect the concept of linear independence. Forming a basis implies a set of vectors that are not only independent but also span the entire vector space, which is a stricter condition. If every vector can be expressed as a combination of others, it signifies dependence rather than independence. While it's true that vectors pointing in different directions can suggest independence, it does not guarantee it mathematically, as they might still lie in the same plane or space. Thus, the condition involving only the trivial solution in the linear combination directly captures the essence of linear independence comprehensively and precisely.