Definition I: Let {v1,v2,…,vn} be a subset of an arbitrary vector space V. If these elements both generate V and are linearly independent
they form a basis for V.
Example I: Show that the vectors (1,1) and (-1,2) form a basis for R2.
Proof: We have to prove that these 2 vectors are both linearly independent and that they generate R2.
Part I: To prove that they are linearly independent, suppose that there are numbers a,b such that:
Part II: To prove that these two vectors generate R2, we have to let (a,b) be an arbitrary element of R2, and show that there exist numbers x,y such that:
Example II: We have already shown that E1, E2,…,En are linearly independent and generate Rn. Therefore, they form a basis for Rn.
Example III: Let W be the vector space generated by et, e2t. We have already shown they are linearly independent. Then they form a basis for W.