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The image shows a vector equation:

\[
\text{El vector } u = (3, 1, 2) \text{ pertenece al } Gen\{(1, -1, 1), (2, 1, 1)\}
\]

This asks whether the vector \( u = (3, 1, 2) \) belongs to the span (denoted by "Gen") of the vectors \( v_1 = (1, -1, 1) \) and \( v_2 = (2, 1, 1) \). To determine this, we need to verify if \( u \) can be expressed as a linear combination of \( v_1 \) and \( v_2 \), i.e., find if there exist scalars \( \alpha \) and \( \beta \) such that:

\[
u = \alpha v_1 + \beta v_2
\]

In vector form, this means:

\[
(3, 1, 2) = \alpha (1, -1, 1) + \beta (2, 1, 1)
\]

This expands to a system of equations:

\[
\begin{aligned}
\alpha + 2\beta &= 3 \\
-\alpha + \beta &= 1 \\
\alpha + \beta &= 2
\end{aligned}
\]

I'll solve this system to find \( \alpha \) and \( \beta \).The system of equations has no solution, meaning that the vector \( u = (3, 1, 2) \) cannot be expressed as a linear combination of the vectors \( (1, -1, 1) \) and \( (2, 1, 1) \). Therefore, the vector \( u \) does not belong to the span of \( \{(1, -1, 1), (2, 1, 1)\} \).

Would you like more details on this process, or do you have any questions?

Here are 5 related questions to deepen your understanding:
1. What does it mean for a vector to belong to the span of other vectors?
2. How can you find the span of multiple vectors?
3. What does it signify when a system of equations has no solution in linear algebra?
4. How do you determine linear dependence between vectors?
5. Can a vector belong to the span of more than two vectors?

**Tip:** When solving a system of linear equations, the absence of solutions indicates the vectors are linearly independent and the given vector is not in the span of the others.

El vector u = (3, 1, 2) pertenece al Gen{(1, -1, 1), (2, 1, 1)}

Explore whether the vector u = (3, 1, 2) belongs to the span of the vectors (1, -1, 1) and (2, 1, 1) by checking if it can be expressed as a linear combination of these vectors. This problem is solved by setting up and solving a system of linear equations, providing insights into linear algebra concepts like vector spaces and linear independence. Suitable for undergraduate students.

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