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The image contains a mathematical exercise in vector spaces. Here is a transcription of the key parts of the text:

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**Tarea no 10 (Clase: Septiembre 3)**

Resolver el problema 2.3 del Capítulo 2 de las notas:

**Determinar cuáles de las siguientes funciones vectoriales son lineales:**

\[
\textbf{a)} \, \overrightarrow{F_1}(r) = \hat{i}r_x + \hat{j}r_y
\]
\[
\textbf{b)} \, \overrightarrow{F_2}(r) = \hat{i}r_x^2 + \hat{j}r_y^2
\]
\[
\textbf{c)} \, \overrightarrow{F_3}(r) = r^2(\hat{i} + \hat{j})
\]

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**El vector \( \hat{k} \) es constante.**

**Sugerencia:**
Poner en cada caso la combinación lineal:
\[
\overrightarrow{r} = \alpha \overrightarrow{p} + \beta \overrightarrow{q}
\]
y efectuar las operaciones indicadas.

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This problem is asking to determine which of the given vector functions are linear.

### To check linearity:
A function \( \mathbf{F}(r) \) is linear if it satisfies the two properties of linearity:
1. **Additivity:** \( \mathbf{F}(r + s) = \mathbf{F}(r) + \mathbf{F}(s) \)
2. **Homogeneity:** \( \mathbf{F}(\lambda r) = \lambda \mathbf{F}(r) \)

Would you like me to walk through each vector function to check if it is linear?

Here are five related questions:
1. Do you know the properties of linear functions in vector spaces?
2. Would you like an explanation of how to apply the linearity conditions?
3. Should I explain how the vector functions \( \hat{i} \) and \( \hat{j} \) are components in 2D space?
4. Are you interested in other examples of linear and non-linear vector functions?
5. Would you like a more detailed explanation of vector operations in this context?

**Tip:** To test if a function is linear, always verify both additivity and homogeneity properties.

Determinar cuáles de las siguientes funciones vectoriales son lineales:
1) F₁(r) = îrₓ + ĵr_y
2) F₂(r) = îrₓ² + ĵr_y²
3) F₃(r) = r²(î + ĵ)

El vector k̂ es constante. Sugerencia: Poner en cada caso la combinación lineal r = αp + βq y efectuar las operaciones indicadas.

Determine the linearity of vector functions F₁, F₂, and F₃ by applying the additivity and homogeneity properties. Explore this advanced linear algebra problem, suitable for undergraduate students.

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