Let's break down the tasks and solve each problem step by step:

1. Determine if the vectors $\mathbf{v_1} = [0, 0, 2]$, $\mathbf{v_2} = [0, 5, 1]$, $\mathbf{v_3} = [2, 1, -8]$ are linearly independent.

To determine linear independence:

• Form a matrix $M$ where each column is one of the vectors: 0 & 0 & 2 \\ 0 & 5 & 1 \\ 2 & 1 & -8 \end{bmatrix}.$$
• Calculate the determinant of $M$. If the determinant is non-zero, the vectors are linearly independent.

2. If $\mathbf{a} = i + 2j$, $\mathbf{b} = k - i$, and $\mathbf{c} = a + 2b$, find $\mathbf{c} \times (\mathbf{a} - \mathbf{b})$.

Steps:

1. Compute $\mathbf{a} - \mathbf{b}$.
2. Compute $\mathbf{c}$ using the given relation.
3. Take the cross product $\mathbf{c} \times (\mathbf{a} - \mathbf{b})$.

3. Find all eigenvalues and corresponding eigenvectors for the matrix $A = \begin{bmatrix} 2 & 1 \\ 4 & 5 \end{bmatrix}$.

Steps:

1. Solve $\det(A - \lambda I) = 0$ to find the eigenvalues ($\lambda$).
2. Substitute each eigenvalue back into $(A - \lambda I)\mathbf{v} = 0$ to find the eigenvectors.

4. Let $\mathbf{v} = (1, 3)$ and $\mathbf{w} = (-4, -2)$. Write $\mathbf{v}$ as the sum of two orthogonal vectors, one of which is the projection of $\mathbf{v}$ onto $\mathbf{w}$.

Steps:

1. Compute the projection of $\mathbf{v}$ onto $\mathbf{w}$: $\text{proj}_{\mathbf{w}} \mathbf{v} = \frac{\mathbf{v} \cdot \mathbf{w}}{\mathbf{w} \cdot \mathbf{w}} \mathbf{w}.$
2. Subtract this projection from $\mathbf{v}$ to find the orthogonal vector: $\mathbf{v} = \text{proj}_{\mathbf{w}} \mathbf{v} + \text{orthogonal component}.$

5. Are the vectors $\mathbf{u} = (8, -4, 1)$, $\mathbf{v} = (3, -2, 5)$, and $\mathbf{w} = (-13, 6, 3)$ in the same plane?

To check if the vectors are coplanar:

• Form a matrix with these vectors as rows or columns: 8 & -4 & 1 \\ 3 & -2 & 5 \\ -13 & 6 & 3 \end{bmatrix}.$$
• Compute the determinant of $M$. If the determinant is zero, the vectors are coplanar.

Would you like me to start solving any of these problems in detail? Let me know!