Proving Linear Map f is a Tensor of Type (1,1)

[rayman123]

Homework Statement

Let V and W be vector spaces and let $$f:V\rightarrow W$$ be a linear map. Show that f is a tensor of type (1,1)

Can someone please show how to do this , I have no idea how to do it.

Homework Equations

The Attempt at a Solution

 

[ystael]

Maybe you should start by explaining what you do know. Can you give the definitions of "vector space", "linear map", "tensor", and "type" of a tensor?

 

[rayman123]

vector space V over a field K is a set in which two operations addition and multiplications by an element of K are definied. The elements of V are called vectors.

Linear map- it is a function between 2 vectors spaces which preserves vector addition and multiplication operations $$f:V\rightarrow W$$
Tensors are objects which map several vectors and dual vectors to a scalar. A tensor of type (p,q) is a multilinear map that maps p dual vectors and q vectors to R.
here we had 2 examples :
a tensor of type (1,0) is a vector but a tensor of type (0,1) maps a vector to a real nr and it is identified with a dual vector.
This is material that our group is going throug now. It is something totally new for us

 

[ystael]

OK, so look at what you have. Obviously $$f$$ is not exactly a tensor of type (1, 1), because it does not map a vector and a dual vector to a scalar; it maps a vector to another vector. The issue is further confused by the fact that $$f: V \to W$$ is a linear map from a space to a different space.

Nevertheless, there is a canonical isomorphism between the space of linear maps from $$V$$ to $$W$$, and the space of tensors that eat a vector in $$V$$ and a dual vector on $$W$$ and return a scalar. This isomorphism is what the question is asking you to find: give a recipe for converting a linear map $$f$$ to such a tensor, and show that this recipe is a one-to-one correspondence.

To see what to do, ask yourself: given $$f(v)$$ for some $$v \in V$$, which is a vector in $$W$$, what information would you need to turn it into a scalar?

 

[rayman123]

hm I am not sure I am following...but to turn something into a scalar we have something like this
$$\omega:V^{*}xVxV\rightarrow R$$ then $$\omega$$ maps dual vector and two vectors into a scalar and is of type (1,2)...but we do not have any dual vector... and tensor maps vecors and dual vectors to a scalar...we have just a map which goes from one vector space to another.

 

[Fredrik]

Use \times for the Cartesian product symbol when you're using LaTeX.

What ystael is saying is that you should be looking for a function T:V×W*→ℝ (or T:W*×V→ℝ). That's a pretty big hint, because it tells you that you can start by writing

T(v,w*)=

where v denotes a member of V, and w* denotes a member of W*, and then try to think of some combination of f, v and w* that makes sense, and is a real number. This combo is what you put on the right-hand side.

I'm surprised that your book calls this a (1,1) tensor. I think it's more common to only consider one vector space V, so that a (1,1) tensor is specifically a map from V×V* into ℝ or from V*×V into ℝ.

 

[rayman123]

Yes I know it is a bit strange, we were thinking about that too, but that is the notation the book follows. Thank you for the replies I will try to do it.

 

[ComradeVVA]

This is Exer. 2.12 from Geometry, Topology, and Physics by Nakahara. He discusses tensors as maps from only one vector space V (and its dual), so it's very strange that all of a sudden we can think of f being a tensor which takes one of its (dual) vectors from another vector space. This is all I could think of as well, however.