Dual Vector Spaces: Understand What They Are

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Dual vector spaces consist of linear functionals that map vectors from a vector space V to its field of scalars. In matrix representation, elements of V are represented as column vectors, while elements of the dual space V* are row vectors, with evaluation performed through matrix multiplication. Understanding dual vector spaces requires both concrete calculations and abstract reasoning, particularly when dealing with tensor representations. A solid grasp of linear algebra is essential for comprehending these concepts in the context of General Relativity. Mastering dual vector spaces will enhance the ability to work with advanced mathematical frameworks in physics.
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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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