91Ó°ÊÓ

In linear algebra, a basis is fundamental for understanding vector spaces. In our exercise, the set \( \mathcal{B} = \{ \cos x, \sin x, x \cos x, x \sin x \} \) forms a basis for the subspace \( W \). A basis is simply a set of vectors that are linearly independent and span the vector space or subspace. In simpler terms, this means:

• No vector in the basis can be written as a combination of the others (independence).
• Any vector in the space can be expressed as a combination of the basis vectors (spanning).

Using \( \mathcal{B} \), we can represent any function in \( W \) as a linear combination of these basis functions. This allows us to solve complex problems by breaking them down into simpler components. Understanding the basis is crucial, as it provides the foundation upon which everything in the vector space is built.

Matrix representation is a powerful tool in linear algebra that translates complex operations into simple matrix computations. In this exercise, we are tasked with representing the differentiation operator \( D \) in terms of our basis \( \mathcal{B} \). The differentiation of each basis element is expressed as a linear combination of the basis itself. The coefficients obtained form the columns of the matrix representation of \( D \). Thus, the matrix \( [D]_{\mathcal{B}} \) depicts how the operator transforms each basis vector.For instance, the differentiation of \( \cos x \) results in \( -\sin x \), captured by the coefficients in the matrix as:\[[D]_{\mathcal{B}} = \begin{bmatrix}0 & 1 & 1 & 0 \ -1 & 0 & 0 & 1 \ 0 & 0 & 0 & 1 \ 0 & 0 & -1 & 0 \end{bmatrix}.\]This matrix simplifies the process of applying the operator to any vector within the subspace, by allowing us to multiply the matrix by the vector's coordinates in the basis to achieve the same transformation as differentiation.

The differentiation operator \( D \) maps functions to their derivatives. In the given context, it transforms each element of the basis, \( \mathcal{B} \), to illustrate how derivatives are expressed within the subspace \( W \).The operation involves calculating the derivative of each basis vector:

• \( \frac{d}{dx}(\cos x) = -\sin x \)
• \( \frac{d}{dx}(\sin x) = \cos x \)
• \( \frac{d}{dx}(x \cos x) = \cos x - x \sin x \)
• \( \frac{d}{dx}(x \sin x) = \sin x + x \cos x \)

By expressing these derivatives as linear combinations of the basis vectors, matrix representation is utilized to expedite calculations. The operator \( D \) is essentially represented by how it modifies these basis functions, which can be instrumental in solving differential equations and understanding the behavior of functions within subspaces.

A subspace in linear algebra is a subset of a vector space that satisfies certain properties. In the exercise, \( W \) is defined as \( \operatorname{span}(\cos x, \sin x, x \cos x, x \sin x) \). This subspace:

• Contains the zero vector.
• Is closed under addition (adding any two vectors in the subspace results in another vector in the subspace).
• Is closed under scalar multiplication (multiplying any vector by a scalar results in another vector in the subspace).

Subspaces are integral to understanding how different components within a vector space relate to each other. They allow us to explore specific characteristics and behaviors of vectors in a contained and simplified manner. Particularly, \( W \) is the space of particular trigonometric polynomial functions, showing how complex structures can be broken down into simpler, more manageable parts.