Problem Breakdown:

The question is asking which of the given polynomials can be added to the set $S = \{1 + 3x, 1 - 6x^2\}$ to form a basis for $P_2$, the space of all polynomials of degree at most 2. A basis for $P_2$ requires:

1. Three linearly independent polynomials (since $\dim(P_2) = 3$).
2. Independence is checked by verifying that the polynomials in the extended set are not linear combinations of each other.

Solution Steps:

1. Represent Polynomials as Vectors:

   • For $S = \{1 + 3x, 1 - 6x^2\}$:

     • $1 + 3x$ corresponds to $[1, 3, 0]$ (constant, $x$, and $x^2$ coefficients).
     • $1 - 6x^2$ corresponds to $[1, 0, -6]$.

   • Additional polynomials:

     • $i: 2 + 3x - 7x^2 \rightarrow [2, 3, -7]$,
     • $ii: 4 + 3x - 19x^2 \rightarrow [4, 3, -19]$,
     • $iii: 11 + 18x - 31x^2 \rightarrow [11, 18, -31]$.

2. Set Up the Matrix: The vectors are rows in the following matrix:

   1 & 3 & 0 \\ 1 & 0 & -6 \\ 2 & 3 & -7 \\ \end{bmatrix}$$ To test if $$i$$ adds independence, check the rank. Repeat the process for $$ii$$ and $$iii$$.

3. Linear Independence Tests: Use row reduction or determinant methods to check whether the rank of the extended matrix increases to 3 when one of the given polynomials is added.

Testing:

Adding Polynomial $i: [2, 3, -7]$: