Linear Systems - Mathematics 214 - Fall 2005

1. Prove that every subspace of Rⁿ has a finite basis.
   Hint:
   Let W be a subspace of Rⁿ.
   We can assume W has a nonzero vector v₁ (why?).
   If W = span(v₁), we're done.
   Otherwise, there is a vector v₂ that is in W but not in span(v₁).
   Show {v₁, v₂} is linearly independent.
   Now repeat: having constructed a linearly independent set of vectors {v₁, ..., v_k}, prove the following:
   If W = span(v₁, ..., v_k), we're done.
   Otherwise, there is a vector v_k+1 that is in W but not in span(v₁, ... v_k).
   Show {v₁, ..., v_k+1} is linearly independent.
   Prove that this process must eventually stop.