If 𝑆 = {𝑣1 , 𝑣2, . . . , 𝑣𝑛 } is a set of vectors in a finite dimensional vector space 𝑉 , then 𝑆 is called a basis for 𝑉 if:

A set of vectors $S = \{ v_1, v_2, \ldots, v_n \}$ in a finite-dimensional vector space $V$ is called a basis for $V$ if it satisfies two important conditions:

1. Linear Independence: The vectors in the set $S$ must be linearly independent. This means that no vector in the set can be expressed as a linear combination of the others. Mathematically, this can be stated as: if $c_1 v_1 + c_2 v_2 + \ldots + c_n v_n = 0$ implies $c_1 = c_2 = \ldots = c_n = 0$, then the vectors are linearly independent.

2. Spanning Set: The set $S$ must span the vector space $V$. This means any vector $v$ in $V$ can be expressed as a linear combination of the vectors in $S$. Specifically, for any vector $v \in V$, there exist scalars $a_1, a_2, \ldots, a_n$ such that $v = a_1 v_1 + a_2 v_2 + \ldots + a_n v_n$.

If both of these conditions are met, then $S$ is considered a basis for the vector space $V$. This foundational concept is crucial in linear algebra as it defines the structure and properties of vector spaces.