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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Answers and Replies
#2
ystael
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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?
#3
rayman123
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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 [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
ystael
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OK, so look at what you have. Obviously [tex]f[/tex] 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 [tex]f: V \to W[/tex] is a linear map from a space to a different space.
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
rayman123
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hm I am not sure I am following....but to turn something into a scalar we have something like this
[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.
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.
#7
rayman123
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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.
#8
ComradeVVA
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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.