91Ó°ÊÓ

A polynomial basis is a set of polynomials that can linearly generate any polynomial within a specified polynomial space. For the polynomial space of degree less than or equal to two, denoted as \( P_2 \), the basis should include polynomials from this space. Each polynomial in \( P_2 \) can be represented as a linear combination of the basis polynomials.
In the original problem, we verified that \( \mathcal{B} = \{1+x^2, 1-x+2x^2, -1-x+kx^2\} \) forms a basis for \( P_2 \). This verification involves checking two key aspects:

• Linear Independence - ensuring no polynomial in the basis can be written as a combination of others.
• Spanning - confirming that any polynomial in \( P_2 \) can be expressed using this basis.

By establishing these criteria, we can affirm that the set \( \mathcal{B} \) successfully acts as a basis for the polynomial space \( P_2 \).

Linear independence is a fundamental concept when determining if a set of vectors, or in this case, polynomials, can serve as a basis for a space. A set of polynomials is linearly independent if the only solution to a linear combination equating to zero involves all coefficients being zero.
In our example with the basis \( \mathcal{B} \), we formulated a system of equations by equating a general linear combination of the basis polynomials to zero. This leads to:

• \( a + b - c = 0 \)
• \( -b - c = 0 \)
• \( a + 2b + kc = 0 \)

We solve this system to confirm that the only solution is \( a = b = c = 0 \). This confirms that no polynomial in the set can be constructed using others, thus proving linear independence. Without linear independence, the set \( \mathcal{B} \) could not serve as a unique basis for \( P_2 \).

The coordinates of a polynomial with respect to a basis are crucial when expressing polynomials as a linear combination of basis elements. For a polynomial \( p(x) \), its coordinates are the coefficients in front of each basis polynomial that reconstruct \( p(x) \) when added together.
For instance, expressing \( p(x) = 1 \) using the basis \( \mathcal{B} = \{1+x^2, 1-x+2x^2, -1-x+kx^2\} \) entails finding the coefficients \( c_1, c_2, c_3 \) such that:
\[ c_1(1+x^2) + c_2(1-x+2x^2) + c_3(-1-x+kx^2) = 1 \] This results in a system of equations whose solution \( (c_1, c_2, c_3) \) gives the coordinates of \( p(x) \). Similarly, any polynomial \( q(x) \) or \( r(x) \) will have unique coordinates with respect to \( \mathcal{B} \), enabling us to efficiently manipulate and understand their relationships using the basis set.

At the heart of polynomial basis, coordinates, and linear independence is the concept of linear combinations. It involves expressing a polynomial as the sum of multiples of other polynomials.
In linear algebra, a linear combination provides a powerful tool for constructing polynomials from a basis. For each polynomial in \( P_2 \), like \( q(x) = 4-2x+7x^2 \), it is expressed as:

• \( q(x) = c_1(1+x^2) + c_2(1-x+2x^2) + c_3(-1-x+kx^2) \)

Finding the coefficients \( c_1, c_2, c_3 \) involves solving systems of equations, allowing us to work within the abstract space \( P_2 \).
Understanding linear combinations opens the door to solving polynomial equations, transforming polynomials, and checking properties like coordinate addition. For example, verifying relationships, such as \[ [4 - 2x + 7x^2]_{\mathcal{B}} + [-2 - 2x + 3x^2]_{\mathcal{B}} \] equiv \[ [2 - 4x + 10x^2]_{\mathcal{B}} \], relies on the mastery of linear combinations.