From the PF rules (https://www.physicsforums.com/showthread.php?t=414380):failexam said:I assigned this one to myself, so technically, it's not a homework problem.
It doesn't matter if the problem is assigned by an instructor or you assign it to yourself.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.
micromass said: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 said: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.
A vector space is a mathematical concept that describes a set of objects, called vectors, that can be added and multiplied by numbers. These operations follow specific rules, such as closure, associativity, and distributivity, and the set must contain a zero vector and have the ability to scale vectors by a scalar.
The adjoint of a vector space is a linear map that describes the relationship between the original vector space and its dual space. It is also known as the transpose in linear algebra and can be thought of as a reflection of the original space across the diagonal.
Vector spaces and their adjoints are closely related because they both describe the same set of objects, but from different perspectives. The vector space describes the original objects, while the adjoint describes the relationships between those objects and their duals. This duality allows us to perform operations on the original space by manipulating its adjoint.
The adjoint plays a crucial role in linear algebra because it allows us to define and manipulate operations on vector spaces in a more efficient and elegant way. It also helps us understand the properties and relationships of vector spaces and their duals, which are essential in many areas of mathematics and physics.
The specific method for calculating the adjoint of a vector space depends on the type of vector space and the operations being performed. In general, the adjoint is found by taking the transpose of a matrix representation of the original space. However, in infinite-dimensional spaces, the adjoint may be defined using integrals or other methods. Familiarity with linear algebra and its properties is crucial in determining the adjoint of a vector space.