Let V=𝒫^(4) denote the space of quartic polynomials, with the L^2 inner product ⟨p, q⟩=∫-1^1 p(x

Let V=𝒫^(4) denote the space of quartic polynomials, with...

Let $V=\mathcal{P}^{(4)}$ denote the space of quartic polynomials, with the $\mathrm{L}^2$ inner product $\langle p, q\rangle=\int_{-1}^1 p(x) q(x) d x$. Let $W=\mathcal{P}^{(2)}$ be the subspace of quadratic polynomials. (a) Write down the conditions that a polynomial $p \in \mathcal{P}^{(4)}$ must satisfy in order to belong to the orthogonal complement $W^{\perp}$. (b) Find a basis for and the dimension of $W^{\perp}$. (c) Find an orthogonal basis for $W^{\perp}$.

Let $V=\mathcal{P}^{(4)}$ denote the space of quartic polynomials, with the $\mathrm{L}^2$ inner product $\langle p, q\rangle=\int_{-1}^1 p(x) q(x) d x$. Let $W=\mathcal{P}^{(2)}$ be the subspace of quadratic polynomials.
(a) Write down the conditions that a polynomial $p \in \mathcal{P}^{(4)}$ must satisfy in order to belong to the orthogonal complement $W^{\perp}$. (b) Find a basis for and the dimension of $W^{\perp}$.
(c) Find an orthogonal basis for $W^{\perp}$.

Key Concepts

Polynomial Function Spaces

Polynomial function spaces like ?? or ?² are vector spaces consisting of all polynomials up to a certain degree. These spaces are finite-dimensional and serve as a fundamental example in linear algebra and approximation theory, where polynomials are used to model functions, solve differential equations, and interpolate data.

Inner Product on Function Spaces

An inner product on a function space, such as the L² inner product defined by an integral, provides a framework for measuring angles and length, enabling the notions of orthogonality and projection. In these spaces, the inner product is used to generalize geometric concepts to infinite-dimensional settings or function spaces.

Orthogonal Complement

The orthogonal complement of a subspace in an inner product space consists of all vectors that are orthogonal to every vector in that subspace. This concept is crucial in various applications, including solving least squares problems, decomposing functions into simpler components, and constructing orthogonal bases.

Basis and Dimension

A basis of a vector space or subspace is a set of linearly independent vectors that span the space, and the number of vectors in the basis is the dimension of the space. Determining a basis is fundamental in linear algebra because it provides a coordinate system for the space, which simplifies calculations and theoretical analysis.

Orthogonal Bases and the Gram-Schmidt Process

An orthogonal basis is a basis in which every pair of distinct vectors is orthogonal under the given inner product, making computations more straightforward. The Gram-Schmidt process is a method for converting any basis into an orthogonal (or orthonormal) basis, and it is widely used in both theoretical investigations and numerical algorithms.

Transcript

00:05 We're given the vector space v of polynomials over r of degree less than or equal to 2.

00:21 So it's p sub 2 of t over r with an inner product product of two functions fmg is the integral from zero to one of ft xtt times g of t bt we have to find a basis of the substates w orthogonal to the function h of two each of t equals two t plus one well, first of all, what song did we send in space? we know that a vexer, i'll call f0, will be orthogonal to h.

01:27 If the inner product of f0 and h equals zero, in other words, zero is equal to i'll let f0 of t2 yeah, anytime we have to google stuff on the show, jingholm well, i let f0 of t be equal to a, t squared plus b t plus c and therefore the inner product of s0 t h is the integral from 0 to 1 a t squared plus b t plus c times 2 t plus 1 you want this to be equal to 0 let's solve this integral this is integral from here to 1 of 2a2a2d2 plus a plus 2b 2b t squared plus b plus c c times t plus c plus c times bc well this is equal to getting a derivative a t to the fourth over two plus a plus two b over three times two q plus b plus 2c over 2, 2 squared, plus c times 2, and t equals 0 to 1, which is a over 2, plus a plus 2 b plus 2 b, over 3, plus b plus 2c over 2 plus c, plus c.

03:41 And this is a over 2 plus a over 3.

03:46 Which is 5 sixth a plus a plus 7 6th 7 6d plus 2c for example, take c equal to zero and take b equal to negative 5, then it follows it, a is equal to positive 7.

04:51 And so we get that f0 of t is 7 t squared minus 5 t.

05:00 You know, azerbaijanis, what the fuck is that shit? it's like an iranian russians.

05:06 Now we know that there's one more vector in the basis for this subspace to find it...

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