https://file.aiquickdraw.com/mathbot/file/1725925795557uvrme4er.jpg

To determine whether the function \( f: \mathbb{R}^3 \to \mathbb{R} \) given by

\[
f \left( \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \right) = x_1^2 + 3x_1x_3 - 4x_2 + x_3^3
\]

is linear, we need to check if it satisfies the two properties of linearity:

1. **Additivity**: \( f(\mathbf{u} + \mathbf{v}) = f(\mathbf{u}) + f(\mathbf{v}) \) for all vectors \( \mathbf{u}, \mathbf{v} \in \mathbb{R}^3 \).
2. **Homogeneity**: \( f(c \mathbf{u}) = c f(\mathbf{u}) \) for all vectors \( \mathbf{u} \in \mathbb{R}^3 \) and scalars \( c \in \mathbb{R} \).

### Analyzing the Function

Let's examine the function:

\[
f(x_1, x_2, x_3) = x_1^2 + 3x_1x_3 - 4x_2 + x_3^3.
\]

A linear function can only have terms that are linear in each variable (i.e., no terms with powers higher than 1 or products of variables). However, in this function:

- The term \( x_1^2 \) is quadratic in \( x_1 \).
- The term \( 3x_1x_3 \) is a product of two variables.
- The term \( x_3^3 \) is cubic in \( x_3 \).

These terms violate the requirement for linearity because a linear function must only have terms that are first-degree monomials (i.e., linear terms like \( ax_1 + bx_2 + cx_3 \)).

### Conclusion

Since the function \( f(x_1, x_2, x_3) = x_1^2 + 3x_1x_3 - 4x_2 + x_3^3 \) contains non-linear terms (such as \( x_1^2 \), \( 3x_1x_3 \), and \( x_3^3 \)), it is **not a linear function**.

Would you like more details on why these properties are necessary for linearity, or have any further questions?

Here are some related questions to consider:
1. What are the properties of a linear function in general?
2. How do you determine if a given transformation is linear?
3. Can a linear function have any constant or quadratic terms?
4. What are some examples of linear functions in \( \mathbb{R}^3 \)?
5. What are the applications of linear functions in real-life scenarios?

**Tip:** To check if a function is linear, always look for non-linear terms such as squared variables or products of variables.

Is the following function on R^3 linear? Given: f([x1, x2, x3]) = x1^2 + 3x1x3 - 4x2 + x3^3

Learn how to determine if a function in R^3 is linear by analyzing the given function: f([x1, x2, x3]) = x1^2 + 3x1x3 - 4x2 + x3^3. This problem covers key concepts in linear algebra, including linearity criteria and function analysis. Suitable for university-level students.

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