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The problem is asking if the vector \((-1, 3, -2)\) is a linear combination of the given vectors:

\[
A = \left\{ (2, 1, 2), (1, 1, 1), (1, 2, -1) \right\}
\]

To determine this, we need to check if there exist real scalars \(\alpha\), \(\beta\), and \(\gamma\) such that:

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

This results in the following system of equations:

\[
\begin{aligned}
2\alpha + \beta + \gamma &= -1 \quad \text{(for the first component)} \\
\alpha + \beta + 2\gamma &= 3 \quad \text{(for the second component)} \\
2\alpha + \beta - \gamma &= -2 \quad \text{(for the third component)}
\end{aligned}
\]

Let me solve this system of equations to find \(\alpha\), \(\beta\), and \(\gamma\).The solution to the system is:

\[
\alpha = -\frac{7}{2}, \quad \beta = \frac{11}{2}, \quad \gamma = \frac{1}{2}
\]

Therefore, the vector \((-1, 3, -2)\) **is** a linear combination of the vectors \((2, 1, 2)\), \((1, 1, 1)\), and \((1, 2, -1)\) with the corresponding scalars \(\alpha = -\frac{7}{2}\), \(\beta = \frac{11}{2}\), and \(\gamma = \frac{1}{2}\).

Would you like further explanation on how to check if vectors are linear combinations? 

Here are some related questions for practice:
1. What is the geometric interpretation of a linear combination of vectors?
2. How can you determine if a set of vectors is linearly independent?
3. What happens if the vector \((-1, 3, -2)\) cannot be written as a linear combination of the given vectors?
4. How do you perform a linear transformation using matrix multiplication?
5. Can you express a general method for solving systems of linear equations like the one above?

**Tip:** Always check the determinant of a matrix formed by vectors to see if they are linearly independent!

Determine si el vector (-1, 3, -2) es una combinación lineal de los siguientes vectores: A = {(2, 1, 2), (1, 1, 1), (1, 2, -1)}. En caso sea cierto, de como respuesta los valores de los escalares reales: α, β, γ.

This problem involves checking if the vector (-1, 3, -2) can be written as a linear combination of the vectors (2, 1, 2), (1, 1, 1), and (1, 2, -1). It requires solving a system of linear equations to find the scalars α, β, and γ.

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