Let W be the plane in R^3 whose general form equation is x + 2y + 3z = 0.
Prove that W is a subspace of R^3. (It's not a valid proof to just say: "It's a subspace b/c it's a plane through the origin." We have never proved such a theorem!)
Find an orthogonal basis for W.
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.
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] in W?
Does the equation Ax=b have a solution? Explain your reasoning.
Find the closest vector b' to b such that the equation Ax=b' has a solution. Hint: Use part (c). Explain your reasoning.
Prove that if u and v are vectors in Rn, then u . proju(v) = u . v
Let W be a subspace of Rn and let v be a vector in Rn. Suppose a, b, c, d are vectors in Rn such that (i) a + b = v = c + d, (ii) a and c are in W, and (iii) b and d are in Wperp. Prove (without looking at the proof in the book) that a = b and c = d.