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The problem asks whether the set \(\{x + 1, x^2 - 1, 2x - 1\}\) is a minimal generating set for the vector space \(P_2 = \{ax^2 + bx + c \mid a, b, c \in \mathbb{R}\}\), which represents the space of polynomials of degree at most 2.

### Step-by-Step Approach:

1. **Dimension of \(P_2\):**
   - The space \(P_2\) consists of all polynomials of the form \(ax^2 + bx + c\), where \(a, b, c \in \mathbb{R}\). This space has dimension 3, because any polynomial in \(P_2\) can be written as a linear combination of the three basis elements \(x^2, x,\) and \(1\).

2. **Check if \(\{x + 1, x^2 - 1, 2x - 1\}\) spans \(P_2\):**
   - A set of vectors spans a space if any element of the space can be written as a linear combination of the vectors in the set.
   - Consider a general polynomial \(ax^2 + bx + c \in P_2\). We need to express this as a linear combination of \(x + 1\), \(x^2 - 1\), and \(2x - 1\). That is, we want to find scalars \(\alpha, \beta, \gamma\) such that:
     \[
     ax^2 + bx + c = \alpha(x + 1) + \beta(x^2 - 1) + \gamma(2x - 1)
     \]
     Expanding the right-hand side:
     \[
     \alpha(x + 1) = \alpha x + \alpha
     \]
     \[
     \beta(x^2 - 1) = \beta x^2 - \beta
     \]
     \[
     \gamma(2x - 1) = 2\gamma x - \gamma
     \]
     Adding these up:
     \[
     \alpha x + \alpha + \beta x^2 - \beta + 2\gamma x - \gamma = \beta x^2 + (\alpha + 2\gamma)x + (\alpha - \beta - \gamma)
     \]
     Now compare this with \(ax^2 + bx + c\). We get the system of equations:
     \[
     \beta = a
     \]
     \[
     \alpha + 2\gamma = b
     \]
     \[
     \alpha - \beta - \gamma = c
     \]

3. **Solve the system:**
   From \(\beta = a\), substitute \(\beta\) into the third equation:
   \[
   \alpha - a - \gamma = c \quad \text{(Equation 3)}
   \]
   Now, solve for \(\alpha\) and \(\gamma\):
   - From Equation 3, \(\alpha = a + \gamma + c\).
   - Substitute this into Equation 2:
     \[
     (a + \gamma + c) + 2\gamma = b
     \]
     Simplify to find \(\gamma\):
     \[
     a + \gamma + c + 2\gamma = b \quad \Rightarrow \quad a + c + 3\gamma = b
     \]
     \[
     \gamma = \frac{b - a - c}{3}
     \]
   Now substitute \(\gamma\) into \(\alpha = a + \gamma + c\):
   \[
   \alpha = a + \frac{b - a - c}{3} + c = \frac{2a + b + 2c}{3}
   \]

Thus, for any polynomial \(ax^2 + bx + c\), we can express it as a linear combination of \(x + 1\), \(x^2 - 1\), and \(2x - 1\). Hence, the set \(\{x + 1, x^2 - 1, 2x - 1\}\) spans \(P_2\).

4. **Minimality:**
   The set \(\{x + 1, x^2 - 1, 2x - 1\}\) contains three elements, which is equal to the dimension of \(P_2\). Therefore, this set is a basis (minimal generating set) for \(P_2\).

### Conclusion:
Yes, \(\{x + 1, x^2 - 1, 2x - 1\}\) is a minimal generating set for \(P_2\).

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### Related Questions:
1. What is the general form of a polynomial in \(P_2\)?
2. How do we determine the dimension of a vector space?
3. How do we verify if a set of vectors spans a vector space?
4. What does it mean for a set of vectors to be linearly independent?
5. How can you find a basis for a polynomial vector space?

### Tip:
When determining if a set is a basis, always check for both spanning and linear independence conditions!

Diketahui P_2 = {ax^2 + bx + c | a, b, c ∈ R} ruang vector atas R. Selidiki apakah {x + 1, x^2 - 1, 2x - 1} merupakan himpunan pembangun minimal untuk P_2.

Explore whether the set {x + 1, x^2 - 1, 2x - 1} forms a minimal generating set for the vector space P_2, which consists of quadratic polynomials. This problem covers concepts in linear algebra such as vector spaces, spanning sets, and basis.

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