a1*v1+a2*v2+...+an*vn=0.
To focus the definition on linear independence, we can say, the vectors v1,v2, ,vn are linearly independent, if and only if the following condition is satisfied:
Whenever a1,a2,...,an are numbers such that:
a1*v1+a2*v2+...+an*vn=0
then ai=0 for all i=1,2,...,n.
Example I:
Show that the vectors (1,1) and (-3,2) in R2 are linearly independent.
Proof:
Let a, b be two numbers such that:
Example II: Let V=Rn and consider the vectors in Rn:
Proof:
Suppose that a1, a2, ,an are elements of
Rn such that
Example III: (Calculus required) Let V be the vector space of all functions of a real variable t. Then the functions et and e2t in V are linearly independent.
Proof:
Then we must prove that whenever there are 2 numbers a and b, such that: