Turn in all of the following problems.

1. The goal 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.
2. 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 point (1,1,1) in R^3. 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 (it's a column vector). Is b in the column space of A? Give a short but rigorous reason without doing any computations! Hint: Is  [1,1,1] 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.