91Ó°ÊÓ

A polynomial vector space is a collection of polynomials that follow specific properties, functioning similarly to vectors in traditional vector spaces. The set of polynomials in a polynomial vector space can vary based on the conditions defining the space. In our exercise, the vector space \( V \) contains polynomials from \( \mathscr{P}_2 \). This indicates polynomials with a degree of at most 2. Each polynomial is of the form \( p(x) = ax^2 + bx + c \). However, a condition is given that \( p(1) = 0 \), which restricts \( V \) such that any polynomial within the space must satisfy this condition.This condition implies that the sum of the coefficients (replacing \( x \) with 1) follows \( a + b + c = 0 \). Consequently, the polynomials are fully characterized by two parameters (since one can be expressed in terms of the others), slightly similar to vectors in a geometric plane.

A basis of a vector space is a collection of vectors that are both linearly independent and span the entire space. These vectors form a kind of 'skeleton' for the vector space. Each polynomial in the vector space can be expressed uniquely as a linear combination of the basis vectors.In the context of polynomial vector spaces, the basis must satisfy any given conditions of the space. For \( V \), the vectors \( \{x^2 - 1, x - 1\} \) were found by rewriting the polynomial after applying the given condition \( a + b + c = 0 \). Expressing this gives us \( p(x) = a(x^2 - 1) + b(x - 1) \).The polynomials \( x^2 - 1 \) and \( x - 1 \) are confirmed as linearly independent because there are no scalar multiples (other than zeros) where one can be written as a linear combination of the other. Therefore, these polynomials form a valid basis for \( V \).

The dimension of a vector space is defined as the number of vectors in its basis. It gives a measure of the "size" or "capacity" of the space in terms of how many directions it can extend.In our problem, the dimension of the vector space \( V \) is determined by the number of polynomials in the basis \( \{x^2 - 1, x - 1\} \). Since there are two polynomials in this basis, the dimension of \( V \) is 2.This tells us that even with the restriction \( p(1) = 0 \), the space maintains two independent "dimensions" or degrees of freedom, meaning it spans a plane-like structure in the space of degree-2 polynomials. Knowing the dimension is crucial as it directly relates to how the space can be manipulated and the types of transformations it supports.