Extra problems

Prove or disprove: If u and v are orthogonal, nonzero vectors in Rⁿ_, then the length of (u+v) is greater than the length of u.

Let W be a subspace of Rⁿ. Let v be a vector in Rⁿ that is not in W. Prove that the length of v is greater than the length of the projection of v onto W. (Hint: use the previous problem.)

Let W be a subspace of Rⁿ, and let v be a vector in Rⁿ. Prove that v is in W iff the length of v is equal to the length of the projection of v onto W. (Compare with Sec 5.2 Problem 27.)

Let W be a subspace of Rⁿ, and let {v₁, ..., v_k} be an orthonormal basis for W. Let u be a vector in Rⁿ. Prove that ||proj_W(u)||² = (u.v₁)² + ... + (u.v_k)² .