91Ó°ÊÓ

In mathematics, a polynomial basis is a set of polynomials that forms a basis for a polynomial space. This means it's a set of polynomials such that any polynomial in the space can be expressed as a linear combination of these basis polynomials. For polynomials of degree at most 3, such as in \( \mathbb{P}_{3} \), we typically choose the standard basis:

• 1
• x
• x^2
• x^3

These are simple, yet powerful tools to understand polynomial spaces. With these polynomials, you can construct any polynomial of degree up to 3 just by scaling and adding them together. This notion is very similar to how we use unit vectors to describe any vector in an ordinary vector space. Understanding the polynomial basis helps us navigate complex problems efficiently.

Polynomials can be thought of as vectors where each coefficient of the polynomial corresponds to a component in a vector. For example, consider a polynomial \(a_0 + a_1 x + a_2 x^2 + a_3 x^3\). This polynomial can be represented as the vector \([a_0, a_1, a_2, a_3]\). When we talk about vectors in polynomial space, we utilize similar concepts for operations as we do in regular vector spaces.
In our exercise, we have the polynomials \( x^3 - 2x^2 - x + 2 \) and \( 3x^3 - x^2 + 2x + 2 \). These can be represented as vectors in polynomial space as follows:

• \( x^3 - 2x^2 - x + 2 \) becomes \([2, -1, -2, 1] \)
• \( 3x^3 - x^2 + 2x + 2 \) turns into \([2, 2, -1, 3] \)

To determine if these vectors are linearly independent, we analyze their coefficients and set up a system of linear equations.

After verifying the linear independence of our given vectors in the polynomial space, the next step is to extend these polynomials to a full basis for \( \mathbb{P}_{3} \). Since our space has a dimension of 4, we need exactly four linearly independent vectors to form a basis.
Our given vectors \( x^3 - 2x^2 - x + 2 \) and \( 3x^3 - x^2 + 2x + 2 \) count as 2 out of the 4 vectors we need. To complete the basis, we can choose two additional polynomials that are guaranteed to be linearly independent from the ones we have. A typical choice are the lowest degree polynomials: \(1\), \(x\), and \(x^2\).
Thus, we can propose:

• \(1\)
• \(x\)

Adding these, we must check for the linear independence among all vectors. For our space \( \mathbb{P}_{3} \), we verify that the combined set \( \{ x^3 - 2x^2 - x + 2, 3x^3 - x^2 + 2x + 2, 1, x \} \) is a linearly independent set, forming the basis for \( \mathbb{P}_{3} \). This method ensures that any polynomial of degree up to 3 can be expressed using these four basis vectors.