Find (a)⟨p, q⟩, (b) p, (c) q, and (d) d(p, q) for the...

Find $(a)\langle p, q\rangle,$ (b) $\|p\|,$ (c) $\|q\|,$ and (d) $d(p, q)$ for the polynomials in $P_{2}$ using the inner product $\langle p, q\rangle= a_{0} b_{0}+a_{1} b_{1}+a_{2} b_{2}$
$$p(x)=1-3 x+x^{2}, \quad q(x)=-x+2 x^{2}$$

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Key Concepts

Representation of Polynomials as Vectors

Polynomials can be represented as vectors by considering their coefficients. For example, a polynomial in P2 (of degree at most 2) can be identified with a vector in R^3. This representation allows for the use of linear algebra techniques such as inner products and norms to study properties of polynomials.

Distance in Inner Product Spaces

The distance between two elements in an inner product space is defined as the norm of their difference. This is analogous to the Euclidean distance between points and provides a measure of how far apart two elements are within the space.

Inner Product in Vector Spaces

An inner product is a function that takes two elements (often considered as vectors) and returns a scalar. It generalizes the dot product in Euclidean space and provides a means of defining geometric concepts such as angles and lengths in more abstract vector spaces, including spaces of polynomials.

Norm Derived from an Inner Product

A norm is a function that assigns a length or magnitude to a vector in a vector space. When the norm is derived from an inner product, it is computed as the square root of the inner product of the vector with itself. This concept extends the idea of distance from Euclidean spaces to any inner product space.

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