Dual Vector Spaces: Understand What They Are

[Ateowa]

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?

 

[Hurkyl]

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

 

[genneth]

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.