Due on Friday, March 29.
For any subspace \(V \subseteq \mathbb{C}^n\), prove that the set \(V^\perp = \{y \in \mathbb{C}^n: \langle y, x \rangle = 0, \forall x \in V\}\) is a subspace (recall that to be a subspace a set needs to be closed under addition and scalar multiplication).
If \(z_1, z_2, \ldots, z_m\) is an orthonormal basis for a subspace \(V \subseteq \mathbb{C}^n\). Let \(P\) be the linear transformation defined by this formula. \[P(x) = \langle x, z_1 \rangle z_1 + \ldots + \langle x, z_m \rangle z_m.\] Prove that the matrix for \(P\) is \(ZZ^*\) where \(Z\) is the \(n\)-by-\(n\) matrix with \(z_1\) through \(z_m\) as the first \(m\) columns, and zeros for all remaining columns.
For the linear transfomation \(P\) above, prove that \(P\) is Hermitian.
Prove that \(P(x)-x\) is orthogonal to \(P(x)\) for all \(x \in \mathbb{C}^n\).
Prove that \(P^2 = P\). Hint: Show that \(P(P(x)) = P(x)\) for all \(x \in \mathbb{C}^n\).