What does it mean for vectors to be linearly independent?

Vectors are considered linearly independent if the only solution to the linear equation formed by those vectors equals zero is the trivial solution, where all scalar coefficients are equal to zero. In the context of the equation c₁v₁ + c₂v₂ + ... + cₖvₖ = 0, if the only values for c₁, c₂, ..., cₖ that satisfy this equation are c₁ = c₂ = ... = cₖ = 0, it implies that none of the vectors can be expressed as a linear combination of the others. This confirms that the vectors do not lie in the same linear space, thereby establishing their independence.

In contrast, if there were a non-trivial solution with at least one coefficient being non-zero, it would indicate that at least one of the vectors could be represented as a combination of others, which would mean they are linearly dependent. This concept is essential in understanding vector spaces, the dimension of those spaces, and the theory behind them in applied linear algebra.