Linear Systems - Mathematics 214 - Fall 2007

Turn in the following problem:

(Recall that the underscore symbol denotes subscripts; e.g., w_1 means w₁.)

1. In this problem we will see why the result of the Gram-Schmidt process is indeed a basis.
   Let W be a subspace of R^n. Suppose {w_1, ..., w_k} is a basis for W (with k >= 2). Let
   w'_1 = w_1,
   w'_2 = w_2 - (projection of w_2 onto span(w'_1)),
   ...
   w'_k = w_k - (projection of w_k onto span(w'_1, ..., w'_{k-1})).
   
   1. Explain why {w'_1, ..., w'_k} is orthogonal. (Hint: You may assume that if u is a vector in R^n and V is a subspace of R^n, then u minus the projection of u onto V is orthogonal to V.)
   2. Explain why neither w'_1 nor w'_2 is zero.
   3. CANCELLED! Explain why none of the vectors w'_1, ..., w'_k is zero.
   4. Explain why {w'_1, ..., w'_k} is linearly independent. (Hint: Use parts a and c.)
   5. Explain why w'_2 is in W.
   6. Explain why w'_3, ..., w'_k are all in W.
   7. Explain why {w'_1, ..., w'_k} is a basis for W.