Dual Vector Spaces: Understand What They Are

In summary, the conversation discusses the topic of dual vector spaces and the confusion surrounding it. The speaker is struggling to understand the concept and has searched online for an explanation. They mention that their professor has lent them a textbook on the topic and they got stuck at the part on dual vector spaces. A suggested explanation is given, which involves using matrix representation for evaluation. The importance of understanding the concept abstractly is also emphasized.
  • #1
Ateowa
25
0
I wasn't quite sure where to post this, as it isn't really a homework question. My professor is teaching us General Relativity from a post-grad book, and I don't have a lot of linear algebra under my belt. He lent me the textbook he's teaching from the other day, and I got stuck when I got to dual vector spaces. I searched online and for some reason I just can't reason out what they are.

Can anyone explain to me what they are?
 
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  • #2
In the matrix representation, elements of V are column vectors and elements of V* are row vectors. Evaluation then becomes the matrix product.
 
  • #3
It might be good to explain what you don't get. Hurkyl's explanation is good -- it's the one that you'll need to do concrete calculations. However, it's important to understand things a bit more abstractly, especially as trying to find matrix representations of tensors tends to tax the imagination.
 

Related to Dual Vector Spaces: Understand What They Are

1. What is a dual vector space?

A dual vector space is the set of all linear functionals on a given vector space. It is a space of linear mappings from the original vector space to the field of scalars.

2. How is a dual vector space related to a vector space?

A dual vector space is the algebraic dual of a vector space, meaning that it is a space of linear mappings from the vector space to the underlying field. It is closely related to the original vector space, as it is constructed from the same set of elements.

3. What is the dimension of a dual vector space?

The dimension of a dual vector space is equal to the dimension of the original vector space. This is because each element in the dual vector space corresponds to a unique linear functional, and the dimension of a space is determined by the number of linearly independent elements.

4. How are elements in a dual vector space represented?

In a finite-dimensional vector space, elements in the dual vector space are represented as row vectors. This is because linear functionals can be represented as linear combinations of the basis vectors of the original vector space.

5. What is the significance of dual vector spaces?

Dual vector spaces play an important role in many areas of mathematics, including linear algebra, functional analysis, and differential geometry. They allow for a deeper understanding of vector spaces and have many applications in areas such as optimization, physics, and computer science.

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