Determine whether or not S = {x² +1, x+ 2, –x² +x} is a basis for P2.
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![### Problem Statement:
Determine whether or not \( S = \{ x^2 + 1, x + 2, -x^2 + x \} \) is a basis for \( P_2 \).
### Solution:
To determine if the set \( S \) forms a basis for \( P_2 \), we need to verify two key properties:
1. **Span**: The set \( S \) must span \( P_2 \), which means any polynomial in \( P_2 \) can be expressed as a linear combination of the polynomials in \( S \).
2. **Linear Independence**: The polynomials in the set \( S \) must be linearly independent, meaning that the only solution to the equation \( c_1 (x^2 + 1) + c_2 (x + 2) + c_3 (-x^2 + x) = 0 \) is \( c_1 = c_2 = c_3 = 0 \).
#### Procedure:
1. **Form the polynomial vector**:
\[
\begin{array}{c}
\left\{ x^2 + 1, \; x + 2, \; -x^2 + x \right\}
\end{array}
\]
2. **Set up the matrix with these polynomials as columns**:
\[
\left[
\begin{matrix}
1 & 0 & -1 \\
0 & 1 & 1 \\
1 & 2 & 0 \\
\end{matrix}
\right]
\]
Each column represents the coefficients of \( x^2 + 1 \), \( x + 2 \), and \( -x^2 + x \) in terms of \( x^2 \), \( x \), and constants, respectively.
3. **Determine the determinant of the matrix** or **row reduce** the matrix to its echelon form. If the matrix is row-reduced to identify its pivot columns, it provides insight into whether the set is linearly independent or not.
After performing row reduction on the matrix:
\[
\left[
\begin{matrix}
1 & 0 & -1 \\
0 & 1 & 1 \\
1 & 2 & 0](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4d548be9-b70c-4345-ad21-fb156d23b920%2Ff316b5ff-b33d-424d-91ae-c34380852a48%2Fupdm6al.png&w=3840&q=75)
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