Let A := (a1, a2, ..., an), where a1,...,an are
vectors in a
vector space V. Then a vector a' in V is said to be
linearly independent of A if it cannot be expressed as a
linear combination of the vectors in A. Alternatively, a vector a' in V is linearly independent of A if it is not in the
span of A.
The system of vectors A is said to be linearly independent if every vector in A is independent of the remaining vectors in A.
Theorem: The system A := (a1, a2, ..., an) is independent if and only if the equation
x1*a1+x2*a2+...+xn*an = 0
Has only one solution, namely, x1=x2=...=xn=0.