Let's analyze the statements in the image and determine which ones are true.

Statements:

1. L'ensemble des combinaisons linéaires de deux vecteurs quelconques de $\mathbb{R}^2$ est $\mathbb{R}^2$
   Translation: The set of linear combinations of two arbitrary vectors in $\mathbb{R}^2$ is $\mathbb{R}^2$.
   Analysis: If you have two linearly independent vectors in $\mathbb{R}^2$, their linear combinations will span $\mathbb{R}^2$. Since this is true for any pair of linearly independent vectors in $\mathbb{R}^2$, this statement is true.

2. L'ensemble $\{(1,2)\}$ engendre une droite passant par l'origine dans $\mathbb{R}^2$
   Translation: The set $\{(1,2)\}$ generates a line passing through the origin in $\mathbb{R}^2$.
   Analysis: A single vector generates a line through the origin in $\mathbb{R}^2$. This statement is true.

3. Si dans un espace vectoriel $V$, tout vecteur est combinaison linéaire de $u$ et $u+v$ et $u+v+w$, alors les vecteurs $\{u, v, w\}$ engendrent $V$
   Translation: If in a vector space $V$, every vector is a linear combination of $u$, $u+v$, and $u+v+w$, then the vectors $\{u, v, w\}$ span $V$.
   Analysis: This statement is false. The fact that every vector is a combination of $u$, $u+v$, and $u+v+w$ does not necessarily imply that $\{u, v, w\}$ are linearly independent and span $V$.

4. L'ensemble $\left\{ \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}, \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \right\}$ engendre $M_{2,2}$
   Translation: The set $\left\{ \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}, \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \right\}$ spans $M_{2,2}$.
   Analysis: $M_{2,2}$ refers to the space of 2x2 matrices, which has dimension 4. The set given contains only 2 matrices, so it cannot span the whole space $M_{2,2}$. This statement is false.

5. L'ensemble $\{(1,0,1), (0,2,3)\}$ engendre $\mathbb{R}^3$
   Translation: The set $\{(1,0,1), (0,2,3)\}$ spans $\mathbb{R}^3$.
   Analysis: $\mathbb{R}^3$ is a 3-dimensional space, but the set only contains 2 vectors. Therefore, it cannot span $\mathbb{R}^3$. This statement is false.

Conclusion:

The correct answer is C. Seulement II et III.

Do you want more details or have any questions?

Related Questions:

1. How do we determine if vectors span a space?
2. What is the dimension of $M_{2,2}$ and why can't two matrices span it?
3. What is the definition of linear independence in vector spaces?
4. How can we verify if a set of vectors in $\mathbb{R}^3$ spans the space?
5. Why do linearly dependent vectors fail to span a vector space?

Tip:

Remember, the number of vectors needed to span a space must be equal to the dimension of that space, and they must be linearly independent.