- #1

wurth_skidder_23

- 39

- 0

Is the following a linear functional?

[tex]\ y (x)=\int_0^1\ t^2 x(t) \, dx [/tex]

[tex]\ y (x)=x(-2)+\int_0^1\ x(t^2)\, dt [/tex]

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- Thread starter wurth_skidder_23
- Start date

In summary, the conversation revolved around understanding linear functionals in the context of a final exam in linear algebra. The individual requested assistance with an example problem and clarification on the definition of a linear functional. Through a series of equations, it was demonstrated that the properties of linearity are satisfied, and thus the given function is indeed a linear functional.

- #1

wurth_skidder_23

- 39

- 0

Is the following a linear functional?

[tex]\ y (x)=\int_0^1\ t^2 x(t) \, dx [/tex]

[tex]\ y (x)=x(-2)+\int_0^1\ x(t^2)\, dt [/tex]

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- #2

StatusX

Homework Helper

- 2,570

- 2

Always start by going back to the definitions.

- #3

wurth_skidder_23

- 39

- 0

For the second one, which is basically just an addition to the first, is this correct?

Property 1 of a linear functional is satisfied as follows:

[tex]\ y (x+z)=x(-2)+z(-2)+\int_0^1\ (x(t^2)+z(t^2))\, dt [/tex]

[tex]\ y (x+z)=x(-2)+\int_0^1\ x(t^2)\, dt + z(-2)+\int_0^1\ z(t^2)\, dt [/tex]

[tex]\ y (x+z)=y(x)+y(z) [/tex]

Property 2 of a linear function is satisfied similarly:

[tex]\ y(a x)=a x(-2)+\int_0^1\ a x(t^2)\, dt [/tex]

[tex]\ y(a x)=a (x(-2)+\int_0^1\ x(t^2)\, dt) [/tex]

[tex]\ y(a x)=a y(x) [/tex]

Property 1 of a linear functional is satisfied as follows:

[tex]\ y (x+z)=x(-2)+z(-2)+\int_0^1\ (x(t^2)+z(t^2))\, dt [/tex]

[tex]\ y (x+z)=x(-2)+\int_0^1\ x(t^2)\, dt + z(-2)+\int_0^1\ z(t^2)\, dt [/tex]

[tex]\ y (x+z)=y(x)+y(z) [/tex]

Property 2 of a linear function is satisfied similarly:

[tex]\ y(a x)=a x(-2)+\int_0^1\ a x(t^2)\, dt [/tex]

[tex]\ y(a x)=a (x(-2)+\int_0^1\ x(t^2)\, dt) [/tex]

[tex]\ y(a x)=a y(x) [/tex]

Last edited:

- #4

StatusX

Homework Helper

- 2,570

- 2

It's as easy as that.

A linear functional is a mathematical function that maps a vector space to its underlying field of scalars. It is a linear transformation that takes a vector as input and outputs a scalar value, such as a real number or complex number.

A linear functional differs from a regular function in that it operates on vectors rather than individual elements. It also follows the properties of linearity, meaning it preserves addition and scalar multiplication.

Understanding linear functionals is important in various fields of mathematics, such as linear algebra and functional analysis. It is also useful in applications such as optimization, signal processing, and physics.

Sure, here is an example problem: Given a linear functional f on a vector space V, find the vector v in V such that f(v) = c, where c is a constant. This is known as the inverse problem for linear functionals.

To improve your understanding of linear functionals, it is helpful to have a strong foundation in linear algebra and functional analysis. You can also practice solving problems and studying different applications of linear functionals. Additionally, seeking out resources such as textbooks and online tutorials can also aid in your understanding.

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