Final answer:
A set of vectors in Rⁿ is linearly independent if the only scalars that make a linear combination of these vectors equal to the zero vector are all zero. This means no vector in the set can be represented as a combination of others, indicating each vector adds a unique dimension to the space.
Step-by-step explanation:
To define whether a set of vectors { a₁ ,…, aₙ }, with each aj being an element of Rⁿ, is linearly independent, one must assess if the only solution to the equation
c₁*a₁ + c₂*a₂ + … + cₙ*aₙ = 0
is when all scalars c₁, c₂, …, cₙ are zero. This equation is known as a linear combination of the vectors, where 0 represents the zero vector. If there are any other solutions where the coefficients cj are not all zero, then the set of vectors is considered to be linearly dependent.
In simpler terms, a set of vectors is linearly independent if no vector in the set can be written as a linear combination of the others. This concept is fundamental in vector spaces and is critical to understanding the span, basis, and dimension of a space in linear algebra.