1. **Problem statement:** 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 $$u = \begin{pmatrix}1 \\ 2 \\ 0 \\ 1\end{pmatrix}, v = \begin{pmatrix}0 \\ 1 \\ 1 \\ 1\end{pmatrix}, w = \begin{pmatrix}1 \\ 0 \\ 2 \\ 0\end{pmatrix}.$$ We need to compute the inner products $\langle u,v \rangle$, $\langle u,w \rangle$, and $\langle v,w \rangle$. 2. **Formula and rules:** The inner product is given by $$\langle x,y \rangle = x^T M y,$$ where $x^T$ is the transpose of vector $x$, and $M$ is a symmetric matrix. To compute this, first multiply $M$ by $y$, then multiply the result by $x^T$. 3. **Compute $\langle u,v \rangle$:** - Calculate $M v$: $$M v = \begin{pmatrix} 2 & -1 & 0 & 0 \\ -1 & 3 & 0 & 0 \\ 0 & 0 & 4 & 1 \\ 0 & 0 & 1 & 2 \end{pmatrix} \begin{pmatrix}0 \\ 1 \\ 1 \\ 1\end{pmatrix} = \begin{pmatrix} 2\cdot0 + (-1)\cdot1 + 0 + 0 \\ -1\cdot0 + 3\cdot1 + 0 + 0 \\ 0 + 0 + 4\cdot1 + 1\cdot1 \\ 0 + 0 + 1\cdot1 + 2\cdot1 \end{pmatrix} = \begin{pmatrix} -1 \\ 3 \\ 5 \\ 3 \end{pmatrix}.$$ - Then compute $u^T (M v)$: $$\langle u,v \rangle = \begin{pmatrix}1 & 2 & 0 & 1\end{pmatrix} \begin{pmatrix} -1 \\ 3 \\ 5 \\ 3 \end{pmatrix} = 1\cdot(-1) + 2\cdot3 + 0 + 1\cdot3 = -1 + 6 + 0 + 3 = 8.$$ 4. **Compute $\langle u,w \rangle$:** - Calculate $M w$: $$M w = \begin{pmatrix} 2 & -1 & 0 & 0 \\ -1 & 3 & 0 & 0 \\ 0 & 0 & 4 & 1 \\ 0 & 0 & 1 & 2 \end{pmatrix} \begin{pmatrix}1 \\ 0 \\ 2 \\ 0\end{pmatrix} = \begin{pmatrix} 2\cdot1 + (-1)\cdot0 + 0 + 0 \\ -1\cdot1 + 3\cdot0 + 0 + 0 \\ 0 + 0 + 4\cdot2 + 1\cdot0 \\ 0 + 0 + 1\cdot2 + 2\cdot0 \end{pmatrix} = \begin{pmatrix} 2 \\ -1 \\ 8 \\ 2 \end{pmatrix}.$$ - Then compute $u^T (M w)$: $$\langle u,w \rangle = \begin{pmatrix}1 & 2 & 0 & 1\end{pmatrix} \begin{pmatrix} 2 \\ -1 \\ 8 \\ 2 \end{pmatrix} = 1\cdot2 + 2\cdot(-1) + 0 + 1\cdot2 = 2 - 2 + 0 + 2 = 2.$$ 5. **Compute $\langle v,w \rangle$:** - Calculate $M w$ (already computed above): $$M w = \begin{pmatrix} 2 \\ -1 \\ 8 \\ 2 \end{pmatrix}.$$ - Then compute $v^T (M w)$: $$\langle v,w \rangle = \begin{pmatrix}0 & 1 & 1 & 1\end{pmatrix} \begin{pmatrix} 2 \\ -1 \\ 8 \\ 2 \end{pmatrix} = 0\cdot2 + 1\cdot(-1) + 1\cdot8 + 1\cdot2 = 0 -1 + 8 + 2 = 9.$$ **Final answers:** $$\langle u,v \rangle = 8, \quad \langle u,w \rangle = 2, \quad \langle v,w \rangle = 9.$$