is a linear transformation and

Is it true that ?

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- Dec 19th 2012, 07:13 AMvercammenVerify the expression.
is a linear transformation and

Is it true that ? - Dec 20th 2012, 01:14 AMchiroRe: Verify the expression.
Hey vercammen.

This might be a dumb question, but is W just a matrix? Also what does raising W to the delta do with regards to the map? - Dec 20th 2012, 04:46 AMvercammenRe: Verify the expression.
As far as I understood U,W are vector fields, S in a k-tensor on W (can be written using basis); S = sum of all possible k-tuples.

- Dec 20th 2012, 12:46 PMchiroRe: Verify the expression.
So is delta an Einstein summation?

- Dec 20th 2012, 02:16 PMvercammenRe: Verify the expression.
just a permutation, I guess

- Dec 20th 2012, 02:25 PMchiroRe: Verify the expression.
The reason I ask is that if it is just a summation, then it should hold (the identity that is).

The reason has to do with distributivity of matrix multiplication, - Dec 20th 2012, 02:35 PMvercammenRe: Verify the expression.
I asked a professor, he told me it's just the matter of using the definitions...

- Dec 22nd 2012, 03:35 PMvercammenRe: Verify the expression.
Here is what I did, but unfortunately it was graded as incorrect.

On the k-tensor powers the induced map is which on pure tensors is

If is a permutation, then and thus we can immediately verify the identity

because both sides will equal

Because pure tensors span the k-tensor power space, we conclude that

(1)

Now let's get back to the problem.

By definition, which is

Now by two consecutive applications of (1), and that is, by definition, - which again by definition equals