Let B be an n x n symmetric matrix such that B² = B. Any such matrix is called a projection matrix (or an orthogonal projection matrix). Given any y in Rⁿ, let ŷ = By and z = y = ŷ.
a. Show that z is orthogonal to ŷ.
b. Let W be the column space of B. Show that y is the sum of a vector in W and a vector in W^⊥?. Why does this prove that By is the orthogonal projection of y onto the column space of B?