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The question you've provided is about vector spaces and basis sets in the context of quadratic polynomials.

**Question Breakdown:**
The polynomial space \( P_2 \) is defined as \( \{ax^2 + bx + c | a, b, c \in \mathbb{R}\} \), which is essentially the space of all quadratic polynomials. The question asks whether the set \( \{x+1, x-1, 2x-1\} \) is a minimal generating set (basis) for \( P_2 \).

To determine if this set is a basis, we need to check two main conditions:
1. **Linear Independence:** No vector in the set can be written as a linear combination of the others.
2. **Spanning the Space:** Every vector in \( P_2 \) can be expressed as a linear combination of vectors in this set.

**Analysis:**
1. **Linear Independence:**
   - We express each vector as a polynomial: \( x+1, x-1, 2x-1 \).
   - Writing these as linear combinations of a standard basis \( \{1, x, x^2\} \), they transform to:
     - \( x+1 = 0x^2 + 1x + 1 \)
     - \( x-1 = 0x^2 + 1x - 1 \)
     - \( 2x-1 = 0x^2 + 2x - 1 \)

   We set up a matrix with these coefficients and perform row reduction to check for independence:

\[
\begin{bmatrix}
0 & 1 & 1 \\
0 & 1 & -1 \\
0 & 2 & -1
\end{bmatrix}
\]

2. **Spanning:** 
   - We need to check if any quadratic polynomial \( ax^2 + bx + c \) can be expressed using these vectors. Given that none of these vectors involve the \( x^2 \) term, they cannot span the entire \( P_2 \) as \( ax^2 \) (where \( a \neq 0 \)) cannot be generated by any combination of \( x+1, x-1, \) and \( 2x-1 \).

**Conclusion:**
The set \( \{x+1, x-1, 2x-1\} \) is not a basis for \( P_2 \) because it does not span the space of all quadratic polynomials, specifically missing any polynomials that include \( x^2 \) terms.

Would you like more details or help with a related topic?

1. How can we modify the set to make it a basis for \( P_2 \)?
2. What does it mean for a set to be linearly independent?
3. What is the dimension of the polynomial space \( P_2 \)?
4. How does the basis of a polynomial space affect its applications in mathematics?
5. Can you explain more about row reduction and how it helps determine linear independence?

**Tip:** Always verify both linear independence and the ability to span the entire vector space when identifying a basis set.

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 of quadratic polynomials P_2. This problem covers key concepts like linear independence, vector spaces, and polynomial basis theory. Suitable for university-level linear algebra courses.

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