91Ó°ÊÓ

A basis for a polynomial space, like any vector space, is a set of vectors—here, polynomials—that are linearly independent and span the entire space. In simpler terms, this means any polynomial of the space can be written as a combination of the basis polynomials. In the exercise, we deal with the polynomial space \( \mathscr{P}_2 \), which consists of all polynomials with a degree of up to 2. Consider the set \( \{1+x, 1+x+x^2\} \). Our goal is to extend this to a basis for \( \mathscr{P}_2 \). To form a complete basis, we need to ensure that every polynomial of degree 2 or less can be expressed as a linear combination of our chosen polynomials. The process involves verifying the linear independence of our existing set and then adding a suitable polynomial to complete the basis. This ensures that together, the polynomials span the entire space.

The dimension of a vector space is the number of vectors in a basis for the space. For polynomial spaces, this counts how many polynomials are needed to express any polynomial in that space. The dimension tells us the `minimum` number of elements required to construct a basis.In \( \mathscr{P}_2 \), the dimension is 3. This is because any polynomial of degree at most 2 can be represented as a linear combination of three terms: a constant, a linear term, and a quadratic term. Typically, these terms are \( 1, x, \) and \( x^2 \). Thus, to form a basis in \( \mathscr{P}_2 \), three linearly independent polynomials are needed.Understanding the space's dimension is crucial as it guides us in completing a partial set of polynomials into a full basis—it tells us how many more polynomials are needed.

Constructing a basis for a polynomial space like \( \mathscr{P}_2 \) involves expanding a given set of polynomials. The exercise starts with the polynomials \( \{1+x, 1+x+x^2\} \). These are already linearly independent, but they do not yet form a complete basis for \( \mathscr{P}_2 \) since only two polynomials are present, yet the space's dimension is 3.To complete the basis, we search for an additional polynomial that complements the existing ones. A natural choice is \( x^2 \), as it's a typical member of any degree-2 polynomial basis. We then verify that adding \( x^2 \) to our set results in all polynomials being linearly independent.Checking the linear independence, we set up the equation: \( a(1+x) + b(1+x+x^2) + c(x^2) = 0 \). Solving this system shows that the only solution is \( a = b = c = 0 \), confirming independence. Therefore, \( \{1+x, 1+x+x^2, x^2\} \) forms a valid basis for \( \mathscr{P}_2 \). This method ensures every polynomial in \( \mathscr{P}_2 \) can be expressed using this set.