1. **Problem (d):** Check if the vector $\mathbf{v} = \begin{bmatrix}1 \\ 1 \\ 1\end{bmatrix}$ is an eigenvector of the matrix $$A = \begin{bmatrix}2 & -1 & 0 \\ 0 & 2 & -1 \\ 2 & 0 & -1\end{bmatrix}$$ and find the corresponding eigenvalue if it is. 2. **Recall:** A vector $\mathbf{v}$ is an eigenvector of $A$ if $$A\mathbf{v} = \lambda \mathbf{v}$$ for some scalar $\lambda$, called the eigenvalue. 3. **Calculate $A\mathbf{v}$:** $$A\mathbf{v} = \begin{bmatrix}2 & -1 & 0 \\ 0 & 2 & -1 \\ 2 & 0 & -1\end{bmatrix} \begin{bmatrix}1 \\ 1 \\ 1\end{bmatrix} = \begin{bmatrix}2(1) -1(1) + 0(1) \\ 0(1) + 2(1) -1(1) \\ 2(1) + 0(1) -1(1)\end{bmatrix} = \begin{bmatrix}1 \\ 1 \\ 1\end{bmatrix}$$ 4. **Compare $A\mathbf{v}$ with $\lambda \mathbf{v}$:** We see that $$A\mathbf{v} = \begin{bmatrix}1 \\ 1 \\ 1\end{bmatrix} = 1 \times \begin{bmatrix}1 \\ 1 \\ 1\end{bmatrix}$$ so $\lambda = 1$. 5. **Conclusion:** $\mathbf{v}$ is an eigenvector of $A$ with eigenvalue $1$. 2. **Problem (e):** Define the norm of a vector in an inner product space and find the norm of the vector $$\mathbf{w} = (i, 1 - i, 1 + i) \in \mathbb{C}^3$$. 3. **Definition:** The norm of a vector $\mathbf{w}$ in an inner product space is $$\|\mathbf{w}\| = \sqrt{\langle \mathbf{w}, \mathbf{w} \rangle}$$ where $\langle \cdot, \cdot \rangle$ is the inner product. 4. **Inner product in $\mathbb{C}^3$:** $$\langle \mathbf{w}, \mathbf{w} \rangle = \sum_{k=1}^3 w_k \overline{w_k}$$ where $\overline{w_k}$ is the complex conjugate of $w_k$. 5. **Calculate:** $$|i|^2 = i \times (-i) = 1$$ $$|1 - i|^2 = (1 - i)(1 + i) = 1 + 1 = 2$$ $$|1 + i|^2 = (1 + i)(1 - i) = 1 + 1 = 2$$ 6. **Sum:** $$1 + 2 + 2 = 5$$ 7. **Norm:** $$\|\mathbf{w}\| = \sqrt{5}$$ 3. **Problem (f):** Find the magnitude of the volume of the box spanned by vectors $$\mathbf{a} = (1,1,1), \mathbf{b} = (1,1,0), \mathbf{c} = (0,1,1)$$. 4. **Recall:** The volume of the parallelepiped spanned by $\mathbf{a}, \mathbf{b}, \mathbf{c}$ is $$V = |\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})|$$ 5. **Calculate $\mathbf{b} \times \mathbf{c}$:** $$\mathbf{b} \times \mathbf{c} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & 1 & 0 \\ 0 & 1 & 1 \end{vmatrix} = \mathbf{i}(1 \times 1 - 0 \times 1) - \mathbf{j}(1 \times 1 - 0 \times 0) + \mathbf{k}(1 \times 1 - 1 \times 0) = \mathbf{i}(1) - \mathbf{j}(1) + \mathbf{k}(1) = (1, -1, 1)$$ 6. **Calculate dot product:** $$\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = (1)(1) + (1)(-1) + (1)(1) = 1 - 1 + 1 = 1$$ 7. **Volume magnitude:** $$V = |1| = 1$$ **Final answers:** - (d) Eigenvalue: $1$ - (e) Norm: $\sqrt{5}$ - (f) Volume magnitude: $1$