The problem presented involves determining the bases of the kernel (ker $f$) and image (im $f$) of the linear transformations (endomorphisms) $f: \mathbb{R}^3 \to \mathbb{R}^3$ and checking if the given transformations are diagonalizable. Here’s how the problem can be approached step by step:

Problem 1: Determine the bases for $\ker f$ and $\text{im }f$ for each case

Example for one of the cases (a):

1. Write the transformation matrix for $f$: For $f(x, y, z) = (2x - y - z, -x + 2y - z, -x - y + 2z)$, represent $f$ in matrix form:

   \begin{bmatrix} 2 & -1 & -1 \\ -1 & 2 & -1 \\ -1 & -1 & 2 \end{bmatrix}$$

2. Find the kernel ($\ker f$): Solve $A \mathbf{v} = 0$, where $\mathbf{v} = (x, y, z)^T$. Perform Gaussian elimination to find the null space of $A$.

3. Find the image ($\text{im }f$): Use the rank-nullity theorem ($\dim(\ker f) + \dim(\text{im }f) = 3$) to determine $\text{im }f$. The rank of $A$ corresponds to the dimension of $\text{im }f$.

4. Provide bases: The solutions for $\ker f$ and $\text{im }f$ give the bases directly.

Problem 2: Investigate diagonalizability

For diagonalizability:

1. Find eigenvalues of $A$: Solve $\det(A - \lambda I) = 0$ for $\lambda$.

2. Find eigenvectors for each eigenvalue: Solve $(A - \lambda I) \mathbf{v} = 0$ to determine the eigenspaces.

3. Check diagonalizability: $A$ is diagonalizable if and only if the dimension of eigenspaces adds up to 3 (the size of $A$).

4. If diagonalizable, construct $P$ and $D$: Use the eigenvectors to form $P$, the diagonalization matrix, and $D = P^{-1}AP$.

Would you like me to solve a specific case (e.g., 1a, 2a) in full detail? If so, please specify!

5 Follow-Up Questions:

1. What are the implications of $\ker f$ and $\text{im }f$ in linear transformations?
2. How does the rank-nullity theorem help simplify computations?
3. What conditions guarantee that an endomorphism is diagonalizable?
4. How do eigenvalues and eigenvectors influence the geometric interpretation of $f$?
5. Would exploring a numerical example of a transformation matrix clarify these concepts?

Tip:

Always verify diagonalizability by checking the total dimension of eigenspaces! This ensures that your computations are consistent with theoretical requirements.