The image presents three mathematical problems related to linear algebra and vector spaces.

Problem Breakdown:

1. First Task:

   • Statement: Let $v = (v_1, \dots, v_p)$ be a system of $p$ vectors in $F$. Show that: $\text{rg}(v) = \text{rg}(m_f(v))$
   • This problem requires proving that the rank of a set of vectors $v$ is equal to the rank of the matrix $m_f(v)$, which could be the matrix representation of the vectors $v$ in the basis $f$ of $F$.

2. Second Task:

   • Statement: Let $u \in \mathcal{L}(E, F)$. Show that: $\text{rg}(u) = \text{rg}(\text{mat}_{e,f}(u))$
   • This problem asks to prove that the rank of the linear map $u$, denoted as $\text{rg}(u)$, is equal to the rank of the matrix representation of $u$ in the bases $e$ and $f$.

3. Third Task (Application):

   • Statement: Consider the map $u: \mathbb{R}^5 \to \mathbb{R}_3[X]$, defined by: $u(a,b,c,d,e) = b + c - d - 3e + (a - c + d + 3e) X + (-a + b + 2c - d) X^2 + (2a + b - c + e) X^3$
   • You are asked to:
     1. Prove that $u$ is a linear map.
     2. Determine its matrix representation with respect to the canonical bases of $\mathbb{R}^5$ and $\mathbb{R}_3[X]$.
     3. Compute the rank and the kernel of $u$.

Strategy to Solve:

1. First Task (Rank of $v$ and its matrix representation):

   • Prove that the rank of a set of vectors remains the same when considering their matrix representation under a specific basis.

2. Second Task (Rank of $u$ and matrix representation):

   • Use properties of linear maps and how their rank corresponds to the rank of their matrix representation in the chosen bases.

3. Third Task (Linear Map from $\mathbb{R}^5$ to $\mathbb{R}_3[X]$):

   • Linearity: You need to verify that $u$ satisfies the properties of a linear map: $u(a + b) = u(a) + u(b)$ and $u(\lambda a) = \lambda u(a)$.
   • Matrix Representation: Write $u(a,b,c,d,e)$ in matrix form by expressing the coefficients of each term in $X^i$ as a linear combination of $a, b, c, d, e$.
   • Rank and Kernel: Compute the rank by finding the number of linearly independent columns in the matrix. The kernel (null space) is determined by solving $u(a,b,c,d,e) = 0$.

Would you like detailed steps for any of these problems? Here are five related questions to explore this topic further:

Related Questions:

1. How do you find the matrix representation of a linear map between two vector spaces?
2. What is the significance of the rank of a linear map in terms of the dimensions of the image and kernel?
3. How can the rank-nullity theorem be used to compute the dimension of the kernel of a linear map?
4. What properties of a set of vectors remain unchanged under a change of basis?
5. How can you determine if a polynomial map (like in problem 3) is injective or surjective?

Tip:

When finding the matrix representation of a linear map, always express each component of the map's output in terms of the chosen basis vectors, then organize these into the matrix form for easy computations of rank and kernel.