Wow! I may mean that, but I have never seen that notation before!

The book is "Introduction to Vectors and Tensors: Volume 1". The authors are Ray M Bowen and C C Wang.

I'm trying to show that an isomorphism exists between

(the set of tensors of order (1,1)) and

(the set of linear maps from V to V).

The first part works quite well. We know that

where

.

Now we have to show that

. To do this let

be a basis for V. Define

linear transformations

by

for

for

If A is an arbitrary member of

then

so

where

.

But we also know that

. This gives that:

We can rearrange this to get:

valid

But we know that A is a linear map so the above statement is valid for any

. ie.

valid

This implies that

meaning that the

linear transformations we defined at the start generate

.

Now we have to prove that these transformations are linearly independent. We do this by setting

.

From here we can use the

linear transformations from before to get:

.

But since

are basis vectors we know that

. Therefore

where

.

Therefore the set of

is a basis of

.

This gives that

.

I find it weird that you went through all this trouble to prove a very elementary fact from linear algebra which, I presume, must be assumed when you reach the necessary level to mess with tensors...
There is also a theorem in the book stating that if two vector spaces have the same dimension then an isomorphism exists between the two. In this case

so an isomorphism exists by the theorem.

"...two vector spaces OVER THE SAME FIELD..."
Finally, I actually want to find an example of an isomorphism between

and

. The book then defines the function

where

and

is an endomorphism of V.

Since the two vector spaces of the same dimensions, by the pigeonhole principle we have to show that

is injective (thus implying that it's a bijection).

The book does this by setting

and

(I thought this was confusing, how does this prove that

is injective?)

Since the scalar product is definite (?) this gives

and thus

.

Consequently the operation "hat" is an isomorphism.

The last part is what i'm particularly confused about. I don't see how setting it to 0 will show that it's injective.

Sorry for the lengthy post, I figured it would be better if I showed exactly what i've done so far!