The problem involves proving that the function $f: \mathbb{R}^3 \to E$, defined by $f(x, y, z) = \begin{pmatrix} x & y & z \\ z & x & y \\ y & z & x \end{pmatrix}$, is an isomorphism and finding the matrix associated with $f$ with respect to specific bases.

Solution:

(i) Montrer que $f$ est un isomorphisme.

To show that $f$ is an isomorphism, we need to prove two properties:

1. Linearity: The function $f$ is linear.
2. Bijectivity: The function $f$ is injective and surjective.

1. Linearity:

$f(x, y, z)$ maps the vector $(x, y, z) \in \mathbb{R}^3$ to a $3 \times 3$ matrix: $f(x, y, z) = \begin{pmatrix} x & y & z \\ z & x & y \\ y & z & x \end{pmatrix}.$ This mapping is linear because:

• $f((x_1, y_1, z_1) + (x_2, y_2, z_2)) = f(x_1 + x_2, y_1 + y_2, z_1 + z_2) = f(x_1, y_1, z_1) + f(x_2, y_2, z_2)$,
• $f(c \cdot (x, y, z)) = c \cdot f(x, y, z)$ for $c \in \mathbb{R}$.

Thus, $f$ is a linear map.

2. Bijectivity:

To check bijectivity, we look at injectivity and surjectivity separately:

• Injectivity: If $f(x, y, z) = 0$ (the zero matrix), then $x = y = z = 0$. Thus, $f$ is injective.

• Surjectivity: The dimension of the image of $f$ is equal to the dimension of $\mathbb{R}^3$, which is 3. Since $f$ is injective and maps to a vector space of the same dimension, it is surjective.

Thus, $f$ is bijective, and hence an isomorphism.

(ii) Déterminer la matrice associée à $f$ par rapport à la base canonique $C$ de $\mathbb{R}^3$ et la base $\mathcal{B}$ de $E$.

We need to compute the matrix of $f$ in the canonical basis $C$ of $\mathbb{R}^3$ and a basis $\mathcal{B}$ of $E$.

1. Let the canonical basis of $\mathbb{R}^3$ be $\{e_1, e_2, e_3\}$, where: $e_1 = (1, 0, 0), \quad e_2 = (0, 1, 0), \quad e_3 = (0, 0, 1).$

2. The images of $e_1, e_2, e_3$ under $f$ are: $f(e_1) = f(1, 0, 0) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix},$ $f(e_2) = f(0, 1, 0) = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix},$ $f(e_3) = f(0, 0, 1) = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}.$

3. If the basis $\mathcal{B}$ of $E$ is formed by the matrices $B_1, B_2, \dots, B_9$, we express $f(e_1), f(e_2), f(e_3)$ as linear combinations of these basis matrices and construct the matrix of $f$.

Let me know if you need a detailed computation of the associated matrix or have any specific questions! Here are some related questions to deepen understanding:

5 Related Questions:

1. What properties must a function satisfy to be an isomorphism in linear algebra?
2. How is the matrix representation of a linear map derived in terms of given bases?
3. What is the importance of bijectivity in proving that a map is an isomorphism?
4. How does one compute the determinant of a linear map, and what role does it play in determining injectivity?
5. What are some common applications of isomorphisms in higher-dimensional linear algebra?

Tip:

When determining if a linear map is an isomorphism, always start by checking the rank of the map and the dimensions of the domain and codomain.