What is V' and its components in mathematics?

[spaghetti3451]

Homework Statement

V' is the adjoint space of the vector space V. For a mathematician, what objects comprise V'?

Homework Equations

The Attempt at a Solution

V' comprises functions, which when applied on the elements of V, produce complex scalars.

 

[micromass]

Is this actually a homework problem?? If it's not, then I'll move it to the general math forums where you can get a complete answer.

Anyway, $V^\prime$ is usually called the dual space of $V$ (the term adjoint is related but is used for something else usually). It consists of all the linear functions of the form $f:V\rightarrow \mathbb{C}$. So all linear functions with codomain $\mathbb{C}$.

 

[spaghetti3451]

I assigned this one to myself, so technically, it's not a homework problem.

Thanks for the answer.

Why do the functions have to be linear? And what is the purpose of defining a dual space of vectors other than to produce complex scalars?

 

[micromass]

The functions have to be linear by definition. We define $V^\prime$ exactly to be linear functions.

I guess what you're asking is why the dual space is useful. That is actually something that is not so easy to answer if you're new to mathematics and physics. It has applications in many fields, including in physics such as quantum mechanics and relativity. But I find it difficult to explain without using advanced math. I guess the dual space is an example of an object that is very useful later on, but whose usefulness in a linear algebra course is not clear at all.

 

[Mark44]

I assigned this one to myself, so technically, it's not a homework problem.

From the PF rules (https://www.physicsforums.com/showthread.php?t=414380):

Any and all high school and undergraduate homework assignments or textbook style exercises for which you are seeking assistance are to be posted in the appropriate forum in our Homework & Coursework Questions area--not in blogs, visitor messages, PMs, or the main technical forums.

It doesn't matter if the problem is assigned by an instructor or you assign it to yourself.

 

[spaghetti3451]

The functions have to be linear by definition. We define $V^\prime$ exactly to be linear functions.

I guess what you're asking is why the dual space is useful. That is actually something that is not so easy to answer if you're new to mathematics and physics. It has applications in many fields, including in physics such as quantum mechanics and relativity. But I find it difficult to explain without using advanced math. I guess the dual space is an example of an object that is very useful later on, but whose usefulness in a linear algebra course is not clear at all.

Thanks for the answer.

 

[spaghetti3451]

From the PF rules (https://www.physicsforums.com/showthread.php?t=414380):
It doesn't matter if the problem is assigned by an instructor or you assign it to yourself.

This definition of what constitutes a homework in Physicsforums should be beneficial to independent learners.