Exercise 5

Let \(B = (e_1, e_2, e_3)\) be the canonical basis of \(\mathbb{R}^3\) and let \(f\) be a linear map defined by:

\[ \begin{cases} f(e_1) = e_1 - e_2 + 3 e_3 \\ f(e_2) = - e_1 + e_2 - 2 e_3 \\ f(e_3) = 2 e_1 + e_2 + e_3 \end{cases} \]

1. Determine the matrix \(M\) of \(f\) in the canonical basis \(B\).
2. Determine the linear map \(f\) associated with \(M\).
3. Let \(B' = (e'_1, e'_2, e'_3)\) with:

   \[
           \begin{cases}
             e'_1 = 2 e_1 + e_2 \\
             e'_2 = 2 e_1 - 2 e_2 \\
             e'_3 = 2 e_3
           \end{cases}
           \]

   Show that \(B'\) is a basis of \(\mathbb{R}^3\).
4. Find the change-of-basis matrix from \(B\) to \(B'\), and then from \(B'\) to \(B\).
5. Let \(X \in \mathbb{R}^3\) be \(X = e_1 - e_3\). Find its coordinates in the basis \(B'\).
6. Determine the matrix \(A\) associated with \(f\) with respect to the basis \(B'\).

Exercise 6

Using Cramer's rule, solve the following linear systems:

1. \[ \begin{cases} 5x - 8y = 1 \\ -7x + 3y = -4 \end{cases} \]
2. \[ \begin{cases} 3x - 2y + z = 0 \\ -2x + y - z = -1 \\ 2x - 4y + 5z = 2 \end{cases} \]
3. \[ \begin{cases} 2i\, x + y = -3 + i \\ 2x + (1 + i) z = 6 \\ (1 - i) y - 6z = 3i \end{cases} \quad \text{with } i^2 = -1 \]