Let V denote the complex vector space spanned by the functions 1, e^i x and e^-i x, where x is a

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Let V denote the complex vector space spanned by the...

Let $V$ denote the complex vector space spanned by the functions $1, e^{\mathrm{i} x}$ and $e^{-\mathrm{i} x}$, where $x$ is a real variable. Which of the following functions belong to $V$ ? (a) $\sin x$, (b) $\cos x-2 \mathrm{i} \sin x$, (c) $\cosh x$, (d) $\sin ^2 \frac{1}{2} x$, (e) $\cos ^2 x ?$

Let $V$ denote the complex vector space spanned by the functions $1, e^{\mathrm{i} x}$ and $e^{-\mathrm{i} x}$, where $x$ is a real variable. Which of the following functions belong to $V$ ?
(a) $\sin x$,
(b) $\cos x-2 \mathrm{i} \sin x$,
(c) $\cosh x$,
(d) $\sin ^2 \frac{1}{2} x$,
(e) $\cos ^2 x ?$

Key Concepts

Hyperbolic Functions

Hyperbolic functions, such as cosh x, are defined in terms of real exponentials and do not directly relate to the imaginary exponential functions used in the basis of the complex vector space. Recognizing the differences in how hyperbolic functions are composed is vital in determining whether they can be represented within the span of 1, e^(i x), and e^(-i x).

Trigonometric Identities

To ascertain whether functions like sin^2(x/2) and cos^2 x belong to the vector space, identities that rewrite these functions in forms involving cos x or sin x or multiple angles are used. These identities enable the expression of squared trigonometric functions in a way that may reveal their dependence on the basis functions given.

Representation of Trigonometric Functions in Exponential Form

Using Euler's formula, trigonometric functions such as sin x and cos x can be written as linear combinations of e^(i x) and e^(-i x). This representation is key to determining if functions involving trigonometric elements are contained in the span of the basis functions.

Euler's Formula

Euler's formula, which states that e^(i x) = cos x + i sin x, is the bridge that connects exponential functions with trigonometric functions. This is central to expressing sine and cosine functions in terms of the given basis elements and establishing whether functions like sin x and combinations of cos x and sin x belong to the space.

Spanning Sets and Basis Functions

A spanning set of functions is a collection from which every function in the vector space can be expressed as a linear combination. In the problem, the functions 1, e^(i x), and e^(-i x) are considered the basis for the space V. Knowing the concept of a span helps determine if other functions, such as trigonometric or hyperbolic functions, can be represented using these basis functions.

Complex Vector Spaces

This concept involves a set of functions that, with the operations of addition and scalar multiplication (with complex scalars), form a vector space. The functions in the space can be combined linearly to create new functions, and understanding the structure of such spaces is crucial for determining whether a given function can be expressed as a linear combination of a specified set of basis functions.

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