Dual Vector Spaces: Understand What They Are

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SUMMARY

Dual vector spaces are fundamental concepts in linear algebra, particularly in the context of General Relativity. In this discussion, participants clarify that elements of a vector space V are represented as column vectors, while elements of the dual space V* are represented as row vectors. The evaluation of these vectors is performed through matrix multiplication. Understanding dual vector spaces is crucial for grasping more complex topics such as tensor representations.

PREREQUISITES
  • Linear algebra fundamentals
  • Matrix representation of vectors
  • Understanding of tensors
  • Basic concepts of General Relativity
NEXT STEPS
  • Study the properties of dual vector spaces in linear algebra
  • Learn about tensor representations and their applications
  • Explore matrix multiplication and its role in vector evaluations
  • Review General Relativity concepts related to vector spaces
USEFUL FOR

Students of mathematics and physics, particularly those studying linear algebra and General Relativity, will benefit from this discussion on dual vector spaces.

Ateowa
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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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In the matrix representation, elements of V are column vectors and elements of V* are row vectors. Evaluation then becomes the matrix product.
 
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.
 

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