Final answer:
Linearly independent vectors v1, v2, and v3 in a vector space V means no vector can be written as a combination of the others, and in three-dimensional space, they span the entire space to define a Cartesian coordinate system.
Step-by-step explanation:
When we say that vectors v₁, v₂, and v₃ are three linearly independent vectors in a vector space V, it means that no vector in this set can be expressed as a linear combination of the others. This implies that the only solution to the equation a₁v₁ + a₂v₂ + a₃v₃ = 0, where a₁, a₂, and a₃ are scalars, is when all these scalars are zero (i.e., a₁ = a₂ = a₃ = 0). In the context of three-dimensional space, this equates to saying that the vectors are not coplanar and span the entire space, allowing for the definition of a Cartesian coordinate system.