1. **Problem 4:** Given vectors $u, v, w$ with inner products $\langle u, v \rangle = 2$, $\langle v, w \rangle = -3$, $\langle u, w \rangle = 5$, and norms $||u||=1$, $||v||=2$, $||w||=7$, find $\langle u - v - 2w, 4u + v \rangle$. 2. **Formula:** The inner product is linear and distributive, so $$\langle a + b, c + d \rangle = \langle a, c \rangle + \langle a, d \rangle + \langle b, c \rangle + \langle b, d \rangle$$ 3. **Step-by-step:** $$\langle u - v - 2w, 4u + v \rangle = \langle u, 4u \rangle + \langle u, v \rangle - \langle v, 4u \rangle - \langle v, v \rangle - 2\langle w, 4u \rangle - 2\langle w, v \rangle$$ 4. Use symmetry $\langle a, b \rangle = \langle b, a \rangle$ and linearity: $$= 4\langle u, u \rangle + \langle u, v \rangle - 4\langle v, u \rangle - \langle v, v \rangle - 8\langle w, u \rangle - 2\langle w, v \rangle$$ 5. Substitute known values: - $\langle u, u \rangle = ||u||^2 = 1^2 = 1$ - $\langle u, v \rangle = 2$ - $\langle v, u \rangle = 2$ - $\langle v, v \rangle = ||v||^2 = 2^2 = 4$ - $\langle w, u \rangle = 5$ - $\langle w, v \rangle = -3$ 6. Calculate: $$= 4(1) + 2 - 4(2) - 4 - 8(5) - 2(-3)$$ $$= 4 + 2 - 8 - 4 - 40 + 6$$ $$= (4 + 2) - 8 - 4 - 40 + 6 = 6 - 8 - 4 - 40 + 6$$ $$= (6 - 8) - 4 - 40 + 6 = -2 - 4 - 40 + 6$$ $$= (-6) - 40 + 6 = -46 + 6 = -40$$ **Answer for problem 4:** $-40$ (option b). --- 7. **Problem 5:** Given matrices $$A = \begin{pmatrix} 2 & -3 \\ 1 & 9 \end{pmatrix}, B = \begin{pmatrix} 0 & 4 \\ -1 & 9 \end{pmatrix}$$ with inner product defined as $$\langle A, B \rangle = u_1 v_1 + u_2 v_2 + u_3 v_3 + u_4 v_4$$ where $u_i, v_i$ are entries of $A, B$ flattened row-wise. 8. **Goal:** Compute $||A + B|| = \sqrt{\langle A + B, A + B \rangle}$. 9. **Calculate $A + B$:** $$A + B = \begin{pmatrix} 2+0 & -3+4 \\ 1+(-1) & 9+9 \end{pmatrix} = \begin{pmatrix} 2 & 1 \\ 0 & 18 \end{pmatrix}$$ 10. **Flatten $A+B$ into vector:** $(2, 1, 0, 18)$. 11. **Compute norm squared:** $$||A + B||^2 = 2^2 + 1^2 + 0^2 + 18^2 = 4 + 1 + 0 + 324 = 329$$ 12. **Norm:** $$||A + B|| = \sqrt{329}$$ **Answer for problem 5:** $\sqrt{329}$ (option B).