- Prove that a set of three vectors in is linearly independent if and only if its span contains the three unit coordinate vectors.
- State and prove a generalization of part (a) for the space .
- Proof. Assume is a basis for . Then are in the span of since they are in and the span of is all of by the definition of basis.
Conversely, assume are in the span of . Let . Since are in the span of we know there are scalars such that
But then if is any vector we have
Then, if for three vectors in then is a basis (the three vectors must be independent, otherwise there would be two vectors that span , contradicting Theorem
- Claim. A set of vectors in is a basis for if and only if its linear span contains the unit coordinate vectors of .
Proof. Assume is a basis for , then . Since the unit coordinate vectors of are in they are then in .
Conversely, assume the unit coordinate vectors are in . Then, by the same argument as in part (a) we know . But, since there are vectors in , we know . Hence, . Furthermore, the vectors of are independent since they span (if they were dependent, then there would be a set with fewer than vectors which would span , contradicting Theorem