Linear Systems - Mathematics 214 - Fall 2007

Turn in all of the following problems.

1. Let W be a subspace of R^n. Let W^perp be the orthogonal complement of W. Prove that the only vector that's in both W and W^perp is the zero vector. (This is proved in the book, but you should do this problem without looking at the book!)
   Hint: Suppose v is a vector in W and W^perp. Explain why v.v must equal zero (use the definition of orthogonal complement). Then conclude that v=0.
2. The purpose of this problem is for you to learn to prove the Orthogonal Decomposition Theorem. So don't look at the proof in the book!
   Let W be a subspace of R^n. Let v be an arbitrary vector in R^n.
   1. Prove that proj_W(v) is orthogonal to perp_W(v). Hints: 1. Take their dot product. 2. Use the definition of perp_W(v).
   2. Prove that v can be written in a unique way as a sum of two vectors one of which is in W, the other in W^perp.
      Hint: Suppose v = w + x and v = w' + x' , where w and w' are in W, and x and x' are in W^perp. Prove that w-w'=x-x'. Now use Problem 1 above to show w-w'=0. Then finish the problem.
3. This problem is a partial review of Sections 5.1 and 5.2 and some older sections.
   Let W be the plane in R^3 whose general form equation is x + 2y + 3z = 0.
   1. Prove that W is a subspace of R^3. (You may not just say: "It's a subspace b/c it's a plane through the origin." We have never proved such a theorem!)
   2. Find an orthogonal basis for W.
   3. Let Q be the head of the vector [1,1,1] in standard position. Find the closest point to Q on W by finding the projection of [1,1,1] onto W.
   4. Let A be the matrix whose columns are the two vectors you found in part (b) above. Let b = [1,1,1]^T (I'm using transpose instead of writing this as a column vector). Is b in the column space of A? Give a short but rigorous reason without doing any computations! Hint: Use the fact that [1,1,1] is not on the plane W.
   5. Does the equation Ax=b have a solution? Explain your reasoning.
   6. Find the closest vector b' to b such that the equation Ax=b' has a solution. Hint: Use part (c). Explain your reasoning.