Chapter 3
Implicit exercise: Lemma 3.2.4. The proof of the second part doesn't seem to use the hypothesis that \(X\) and \(Y\)are independent, but clearly the conclusion does not hold unless they are independent. Where is the hypothesis of independence actually used?
\(\mathbf{E}\left\langle X,Y\right\rangle^2=\mathbf{E\mathbf{_YE_{X\left|Y\right|}}}\left(\left\langle X,Y\right\rangle^2\left|Y\right|\right)\) is what the law of total expectation states in general. We can replace the inner conditional expectation with an unconditional expectation \(E_X\) only because of the independence.
Chapter 4
Exercise 4.1.1 Since the matrix \(A\) is invertible \(A^{-1}\) is unique. Therefore, we can multiply \(A\) by the proposed expression for \(A^{-1}\) and confirm that we obtain \(I_n\) because the \(u_i\) and \(v_i\) are both orthonormal bases of \(\mathbf{R}^n\).