The image shows a mathematical question asking which of the given statements are true or false. The problem is about vector spaces, subspaces, and linear independence in various mathematical contexts. Here's a breakdown of each statement and the correctness (in blue) marked in the image:

Statement Breakdown:

1. Statement I: "The set of vectors $\{(1,2), (2,1), (0,1)\}$ spans $\mathbb{R}^2$."
   Marked as True (Vrai): This is correct. These three vectors are linearly independent and span $\mathbb{R}^2$, so the set generates $\mathbb{R}^2$.

2. Statement II: "The set $\{(x, y, z) \mid x - y + 2z = -1\}$ is a subspace of $\mathbb{R}^3$."
   Marked as False (Faux): This is correct. A subspace of $\mathbb{R}^3$ must contain the zero vector, but this equation does not pass through the origin (since for $x = y = z = 0$, the equation gives $-1$), so it is not a subspace.

3. Statement III: "In a vector space $V$, if there exists a vector that is a linear combination of $u - v$ and $u + w$, then the vectors $\{u, v, w\}$ span $V$."
   Marked as False (Faux): This is correct. The linear combination condition provided does not necessarily imply that $\{u, v, w\}$ will span the space $V$, as more specific conditions are needed for this.

4. Statement IV: "The set $\{p(x) \in \mathbb{P}_2 \mid p(-1) = 0\}$ is a subspace of $\mathbb{P}_2$, where $\mathbb{P}_2$ is the space of polynomials of degree at most 2."
   Marked as True (Vrai): This is correct. The condition $p(-1) = 0$ is a homogeneous condition, and the set of such polynomials forms a subspace of $\mathbb{P}_2$.

5. Statement V: "The set $\{\cos^2(3x), \sin^2(3x), 2\}$ is linearly dependent in $\mathcal{F}(\mathbb{R})$, where $\mathcal{F}(\mathbb{R})$ is the space of functions defined on $\mathbb{R}$."
   Marked as True (Vrai): This is correct. The trigonometric identity $\cos^2(3x) + \sin^2(3x) = 1$ shows that these functions are linearly dependent.

Would you like further details or explanations on any of these topics?

Five Related Questions:

1. What are the conditions for a set of vectors to span a vector space?
2. How do you verify if a set of functions is linearly dependent?
3. What is the significance of the zero vector in determining if a set is a subspace?
4. Can you explain why the condition $p(-1) = 0$ ensures a subspace of $\mathbb{P}_2$?
5. How do trigonometric identities help in establishing linear dependence of functions?

Tip:

When checking whether a set is a subspace, always verify that it contains the zero vector and is closed under addition and scalar multiplication.