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Let's analyze the two questions in the image:

### Question 3: Is the following function on \(\mathbb{R}^3\) linear?
Given function:
\[
f\left( \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \right) = x_1^2 + 3x_1x_3 - 4x_2 + x_3^3
\]
We are asked to determine if this function is linear.

**Answer: No, the function is not linear.**

#### Explanation:
For a function to be linear, it must satisfy two conditions:
1. **Additivity**: \( f(\mathbf{x} + \mathbf{y}) = f(\mathbf{x}) + f(\mathbf{y}) \)
2. **Homogeneity**: \( f(c\mathbf{x}) = c f(\mathbf{x}) \)

However, the given function contains terms like \( x_1^2 \) and \( x_3^3 \), which are non-linear. Specifically, the powers of \(x_1\) and \(x_3\) are not 1, so the function does not satisfy the criteria for linearity.

### Question 4: Prove that every real-valued linear function on \(\mathbb{R}^n\) is of the form
\[
f\left( \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix} \right) = a_1x_1 + a_2x_2 + \cdots + a_nx_n
\]
where \(a_1, a_2, \dots, a_n \in \mathbb{R}\) are some numbers.

#### Hint: \( a_j = f(e_j) \), where \( e_j \) is the standard basis vector.

#### Proof Outline:
1. Let \(f\) be a real-valued linear function on \(\mathbb{R}^n\). This means \(f\) is linear, and thus must satisfy additivity and homogeneity.

2. The function \(f\) can be completely determined by its action on the standard basis vectors \(e_1, e_2, \dots, e_n\), where \( e_j \) is a vector with 1 in the \(j\)-th position and 0 elsewhere.

3. Since \(f\) is linear, we can express the function \(f\) for any vector \( \mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix} \) as a linear combination of the \(x_j\)'s:
\[
f(\mathbf{x}) = f\left( \sum_{j=1}^{n} x_j e_j \right)
\]
4. By the linearity of \(f\), we can distribute the function across the sum:
\[
f(\mathbf{x}) = \sum_{j=1}^{n} x_j f(e_j)
\]
5. Let \( a_j = f(e_j) \), as given in the hint. Then:
\[
f(\mathbf{x}) = \sum_{j=1}^{n} a_j x_j
\]
This is exactly the desired form.

Thus, every real-valued linear function on \(\mathbb{R}^n\) can be written as:
\[
f(\mathbf{x}) = a_1x_1 + a_2x_2 + \cdots + a_nx_n
\]

Would you like to go over any part of this in more detail? 

### Related Questions:
1. What are the conditions for a function to be considered linear in \(\mathbb{R}^n\)?
2. How does the structure of non-linear terms affect linearity?
3. What are some practical examples of linear functions in physics or economics?
4. How do we construct standard basis vectors in \(\mathbb{R}^n\)?
5. How can we use matrix transformations to represent linear functions in higher dimensions?

#### Tip:
Always check if terms have powers greater than 1 or are products of variables to quickly identify non-linearity in a function.

3) Is the following function on R^3 linear? f([x1, x2, x3]^T) = x1^2 + 3x1x3 - 4x2 + x3^3
4) Prove that every real-valued linear function on R^n is of the form f([x1, x2, ..., xn]^T) = a1x1 + a2x2 + ... + anxn, where a1, ..., an ∈ R are some numbers.

Explore the conditions that define linearity for functions on R^3 and prove the form of every real-valued linear function on R^n. This page provides a comprehensive solution suitable for undergraduate students.

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