91Ó°ÊÓ

A coordinate vector refers to a representation of a polynomial in terms of the coefficients that express it as a linear combination of basis vectors. In our example, the polynomial \( \mathbf{p}(t) = 3 + t - 6t^2 \) is expressed in terms of the basis \( \mathcal{B} = \{1-t^2, t-t^2, 2-2t+t^2\} \). This means we are looking to find coefficients \( c_1, c_2, \) and \( c_3 \) such that the polynomial can be written as a combination of these bases.

• The coordinate vector of \( \mathbf{p}(t) \) relative to \( \mathcal{B} \) will be \([7, -3, -2]\).
• This vector uniquely represents \( \mathbf{p}(t) \) within the vector space \( \mathbb{P}_2 \).

Understanding coordinate vectors is essential for working with polynomial expressions and changing between different bases, making them a fundamental concept in linear algebra and vector spaces.

Polynomial expressions are mathematical expressions consisting of variables (such as \( t \)), coefficients, and the operations of addition, subtraction, and multiplication. A polynomial of degree \( n \) has the general form \( a_n t^n + a_{n-1} t^{n-1} + \dots + a_1 t + a_0 \).

• In our exercise, we dealt with polynomials of degree at most 2, i.e., quadratic polynomials.
• The polynomial \( \mathbf{p}(t) = 3 + t - 6t^2 \) is a quadratic polynomial.

Polynomials can be manipulated and transformed by expressing them in terms of different bases, such as in the method for finding a coordinate vector.

In linear algebra, a basis for a vector space is a set of vectors such that:- They span the vector space, meaning any vector in the space can be expressed as a linear combination of these basis vectors.- They are linearly independent, meaning no vector in the set can be represented as a combination of the others.In our exercise, the set \( \mathcal{B} = \{1-t^2, t-t^2, 2-2t+t^2\} \) forms a basis for \( \mathbb{P}_2 \), the vector space of polynomials with degree at most 2.

• Each polynomial in the basis is unique and cannot be formed by combining the others.
• Any polynomial in \( \mathbb{P}_2 \) can be expressed as a linear combination of the polynomials in \( \mathcal{B} \).

Choosing a basis is an important concept because it allows for a structured way to represent and manipulate vectors within a space. It essentially provides a 'coordinate system' for understanding vectors, similar to the way coordinates work in geometry.