Problem 1.
Let W be a subspace of $\mathbb{R}^n$ and let $\{u_1, \dots, u_m\}$ be an orthonormal basis of W.
(1) Show that for $v \in \mathbb{R}^n$, we have $||\alpha v|| = |\alpha| ||v||$ for $\alpha \in \mathbb{R}$.
(2) Show that
$||w||^2 = |(w, u_1)|^2 + \dots + |(w, u_m)|^2$
for $w \in W$. Hint: as shown in class, $w = (w, u_1)u_1 + \dots + (w, u_m)u_m$.
Use the Pythagorean theorem and (1).
Problem 2. Axler 6.20
Let W be a subspace of $\mathbb{R}^n$, let $P = P_W$ be the orthogonal projection onto W, and
let $A \in M_n(\mathbb{R})$. Show that the following are equivalent:
(1) W and $W^\perp$ are both A-invariant
(2) $PA = AP$
Hints:
• $Pw = w$ for $w \in W$
• $Py = 0$ for $y \in W^\perp$
• for $v \in \mathbb{R}^n$, you can write $v = \hat{v} + y$ with $\hat{v} \in W$ and $y \in W^\perp$
