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The problem is asking to write the matrix \( M \) of the linear transformation \( f \) in the canonical basis \( \mathcal{B} = (e_1, e_2, e_3) \) of \( \mathbb{R}^3 \), where \( f \) is defined by its action on the basis vectors:

\[
f(e_1) = -3e_2 + 3e_3
\]
\[
f(e_2) = -4e_1 - 4e_2 + 4e_3
\]
\[
f(e_3) = 2e_1 - e_2 + e_3
\]

The goal is to express the matrix representation \( M \) of \( f \) with respect to the canonical basis. 

### Step-by-Step:
The matrix \( M \) is constructed by placing the coordinates of \( f(e_1) \), \( f(e_2) \), and \( f(e_3) \) as columns.

1. **First Column**: The coordinates of \( f(e_1) \) in terms of \( e_1, e_2, e_3 \) are:
   \[
   f(e_1) = -3e_2 + 3e_3 = \begin{pmatrix} 0 \\ -3 \\ 3 \end{pmatrix}
   \]

2. **Second Column**: The coordinates of \( f(e_2) \) are:
   \[
   f(e_2) = -4e_1 - 4e_2 + 4e_3 = \begin{pmatrix} -4 \\ -4 \\ 4 \end{pmatrix}
   \]

3. **Third Column**: The coordinates of \( f(e_3) \) are:
   \[
   f(e_3) = 2e_1 - e_2 + e_3 = \begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}
   \]

### Matrix Representation:
Thus, the matrix \( M \) of \( f \) in the canonical basis \( \mathcal{B} \) is:
\[
M = \begin{pmatrix} 
0 & -4 & 2 \\
-3 & -4 & -1 \\
3 & 4 & 1
\end{pmatrix}
\]

This matrix describes the linear transformation \( f \) with respect to the canonical basis of \( \mathbb{R}^3 \).

Would you like further details or explanations on any step?

Here are 5 related questions to consider:
1. How can you verify if a matrix represents a linear transformation?
2. What is the significance of the canonical basis in this context?
3. How would the matrix change if we used a different basis?
4. Can this matrix be inverted? If so, how would you find the inverse?
5. What are the eigenvalues and eigenvectors of this matrix?

**Tip**: When constructing a matrix for a linear transformation, always express the transformation in terms of the basis vectors before building the columns.

Write the matrix M of the linear transformation f in the canonical basis B = (e1, e2, e3) of R^3, where f is defined as follows: f(e1) = -3e2 + 3e3, f(e2) = -4e1 - 4e2 + 4e3, and f(e3) = 2e1 - e2 + e3.

Learn how to write the matrix M of the linear transformation f in the canonical basis B = (e1, e2, e3) of R^3. This involves constructing a matrix from the transformations of the standard basis vectors e1, e2, and e3, with a step-by-step solution. Suitable for linear algebra students.

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