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## Homework Statement

Let V and W be vector spaces and let [tex] f:V\rightarrow W[/tex] 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.

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- Thread starter rayman123
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- #1

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Let V and W be vector spaces and let [tex] f:V\rightarrow W[/tex] 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.

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- #3

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Linear map- it is a function between 2 vectors spaces which preserves vector addition and multiplication operations [tex] f:V\rightarrow W[/tex]

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

- #4

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

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

- #5

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[tex] \omega:V^{*}xVxV\rightarrow R[/tex] then [tex] \omega[/tex] 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.

- #6

Fredrik

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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.

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