1. **State the problem:** We are given an inner product on $\mathbb{R}^4$ defined by $\langle x,y \rangle = x^T M y$ where $$M = \begin{pmatrix} 2 & -1 & 0 & 0 \\ -1 & 3 & 0 & 0 \\ 0 & 0 & 4 & 1 \\ 0 & 0 & 1 & 2 \end{pmatrix}$$ and vectors $$u = \begin{pmatrix}1 \\ 2 \\ 0 \\ 1\end{pmatrix}, \quad v = \begin{pmatrix}0 \\ 1 \\ 1 \\ 1\end{pmatrix}, \quad w = \begin{pmatrix}1 \\ 0 \\ 2 \\ 0\end{pmatrix}.$$ We want to compute the norms $\|u\|$, $\|v\|$, and $\|w\|$ induced by this inner product. 2. **Recall the norm formula:** The norm induced by an inner product is $$\|x\| = \sqrt{\langle x, x \rangle} = \sqrt{x^T M x}.$$ 3. **Calculate $\|u\|$:** $$M u = \begin{pmatrix} 2 & -1 & 0 & 0 \\ -1 & 3 & 0 & 0 \\ 0 & 0 & 4 & 1 \\ 0 & 0 & 1 & 2 \end{pmatrix} \begin{pmatrix}1 \\ 2 \\ 0 \\ 1\end{pmatrix} = \begin{pmatrix} 2(1) -1(2) + 0 + 0 \\ -1(1) + 3(2) + 0 + 0 \\ 0 + 0 + 4(0) + 1(1) \\ 0 + 0 + 1(0) + 2(1) \end{pmatrix} = \begin{pmatrix} 2 - 2 \\ -1 + 6 \\ 1 \\ 2 \end{pmatrix} = \begin{pmatrix}0 \\ 5 \\ 1 \\ 2 \end{pmatrix}.$$ Then $$u^T M u = u^T (M u) = \begin{pmatrix}1 & 2 & 0 & 1\end{pmatrix} \begin{pmatrix}0 \\ 5 \\ 1 \\ 2 \end{pmatrix} = 1(0) + 2(5) + 0(1) + 1(2) = 0 + 10 + 0 + 2 = 12.$$ So $$\|u\| = \sqrt{12} = 2\sqrt{3}.$$ 4. **Calculate $\|v\|$:** $$M v = \begin{pmatrix} 2 & -1 & 0 & 0 \\ -1 & 3 & 0 & 0 \\ 0 & 0 & 4 & 1 \\ 0 & 0 & 1 & 2 \end{pmatrix} \begin{pmatrix}0 \\ 1 \\ 1 \\ 1\end{pmatrix} = \begin{pmatrix} 2(0) -1(1) + 0 + 0 \\ -1(0) + 3(1) + 0 + 0 \\ 0 + 0 + 4(1) + 1(1) \\ 0 + 0 + 1(1) + 2(1) \end{pmatrix} = \begin{pmatrix} -1 \\ 3 \\ 5 \\ 3 \end{pmatrix}.$$ Then $$v^T M v = \begin{pmatrix}0 & 1 & 1 & 1\end{pmatrix} \begin{pmatrix}-1 \\ 3 \\ 5 \\ 3 \end{pmatrix} = 0(-1) + 1(3) + 1(5) + 1(3) = 0 + 3 + 5 + 3 = 11.$$ So $$\|v\| = \sqrt{11}.$$ 5. **Calculate $\|w\|$:** $$M w = \begin{pmatrix} 2 & -1 & 0 & 0 \\ -1 & 3 & 0 & 0 \\ 0 & 0 & 4 & 1 \\ 0 & 0 & 1 & 2 \end{pmatrix} \begin{pmatrix}1 \\ 0 \\ 2 \\ 0\end{pmatrix} = \begin{pmatrix} 2(1) -1(0) + 0 + 0 \\ -1(1) + 3(0) + 0 + 0 \\ 0 + 0 + 4(2) + 1(0) \\ 0 + 0 + 1(2) + 2(0) \end{pmatrix} = \begin{pmatrix} 2 \\ -1 \\ 8 \\ 2 \end{pmatrix}.$$ Then $$w^T M w = \begin{pmatrix}1 & 0 & 2 & 0\end{pmatrix} \begin{pmatrix}2 \\ -1 \\ 8 \\ 2 \end{pmatrix} = 1(2) + 0(-1) + 2(8) + 0(2) = 2 + 0 + 16 + 0 = 18.$$ So $$\|w\| = \sqrt{18} = 3\sqrt{2}.$$ **Final answers:** $$\|u\| = 2\sqrt{3}, \quad \|v\| = \sqrt{11}, \quad \|w\| = 3\sqrt{2}.$$