3 ms·

And how do you define exterior powers without using a basis?

Sniffnoy 9y ago
The definition of exterior power doesn't rely on a basis. The k'th exterior power is the quotient of the k'th tensor power by things of the form (v_1 ⊗ v_2 ... ⊗ v_k), where two of the v_i are equal. I'm assuming you know how to define a tensor product without resort to bases. If not, see: https://en.wikipedia.org/wiki/Tensor_product#Definition https://en.wikipedia.org/wiki/Tensor_product#Definition (it's not the clearest exposition of it, but it'll do) (Or, if you like, you could define the tensor algebra, take a quotient of that to get the exterior algebra, and then restrict to the image of the k'th tensor power to get the k'th exterior power.) Again, obviously you need to use bases to prove how to compute the dimension of an exterior power. But you don't need them just to define it.
- mathgenius 9y ago
  Right. It's kind of interesting how in all of these definitions the "easy" way involves using a basis. (Although we could argue about what the "easy" way is.)
- mathgenius 9y ago
  I thought a bit more about this. Strictly speaking the construction of the tensor product that you link to does use a basis. This basis is the Product of the original two vector spaces. Also, the relations that you quotient by, this is a basis for a subspace. We didn't need a basis for the original spaces, but ended up using bases elsewhere.
  - Sniffnoy 9y ago
    That's certainly true, but that's also clearly not what was being talked about. The problem was (easier) avoiding choosing bases to perform a construction, and (harder) avoiding assuming all vector spaces have bases. Using bases that you directly contruct isn't something anyone ever really has reason to avoid. (And really, neither is the second in the finite-dimensional case, but hey, may as well if you can, right?)
  - zodiac 9y ago
    You can define tensor products and quotient spaces without bases! (It's a bit of work though)
  - jacobolus 9y ago
    http://geocalc.clas.asu.edu/pdf/MathViruses.pdf http://geocalc.clas.asu.edu/pdf/MathViruses.pdf

The definition of exterior power doesn't rely on a basis. The k'th exterior power is the quotient of the k'th tensor power by things of the form (v_1 ⊗ v_2 ... ⊗ v_k), where two of the v_i are equal. I'm assuming you know how to define a tensor product without resort to bases. If not, see: https://en.wikipedia.org/wiki/Tensor_product#Definition https://en.wikipedia.org/wiki/Tensor_product#Definition (it's not the clearest exposition of it, but it'll do) (Or, if you like, you could define the tensor algebra, take a quotient of that to get the exterior algebra, and then restrict to the image of the k'th tensor power to get the k'th exterior power.) Again, obviously you need to use bases to prove how to compute the dimension of an exterior power. But you don't need them just to define it.

mathgenius 9y ago
Right. It's kind of interesting how in all of these definitions the "easy" way involves using a basis. (Although we could argue about what the "easy" way is.)
mathgenius 9y ago
I thought a bit more about this. Strictly speaking the construction of the tensor product that you link to does use a basis. This basis is the Product of the original two vector spaces. Also, the relations that you quotient by, this is a basis for a subspace. We didn't need a basis for the original spaces, but ended up using bases elsewhere.
- Sniffnoy 9y ago
  That's certainly true, but that's also clearly not what was being talked about. The problem was (easier) avoiding choosing bases to perform a construction, and (harder) avoiding assuming all vector spaces have bases. Using bases that you directly contruct isn't something anyone ever really has reason to avoid. (And really, neither is the second in the finite-dimensional case, but hey, may as well if you can, right?)
- zodiac 9y ago
  You can define tensor products and quotient spaces without bases! (It's a bit of work though)
- jacobolus 9y ago
  http://geocalc.clas.asu.edu/pdf/MathViruses.pdf http://geocalc.clas.asu.edu/pdf/MathViruses.pdf