Tensor homework.
Renzo’s math 469
Decomposable tensors Let V and W be two dimensional vector spaces
with bases e1 , e2 , f1 , f2 , respectively. Prove that the tensor
T = a e1 ⊗ f1 + b e1 ⊗ f2 + c e2 ⊗ f1 + d e2 ⊗ f2
is decomposable if and only if ad − bc = 0. Now replace V with V ∗ ,
so that you can think of T as a linear transformation from V to W .
What does T being decomposable correspond to in the language of
linear functions. Why?
Changing basis .
1. Let V have
bases e1 , . . . , en and f1 , . . . , fn , and for each j
P two
i
let fj = i aj ei . The tensor A = aij is called the matrix of base
change for V . Given a vector
X
v=
v k fk ,
how do you describe the vector v in the basis e1 , . . . , en ?
P
2. Let f k = i ãkl el give the transformation for the dual vectors.
Prove that the matrix ãkl is A−1 .
3. Given a general operator T ∈ L(V ), described as a tensor in some
basis e1 , . . . , en , describe how the tensor changes when changing
to a different basis f1 , . . . , fn .
The identity function .
1. Show that the identity operator
in Id. ∈ L(V ) is expressed, in
P i
any basis, by the tensor
e ⊗ ei .
2. Show that any linear operator T ∈ L(V, W ) is seen as a tensor as
follows:
• Consider the map IdV ∗ ⊗ T : V ∗ ⊗ V → V ∗ ⊗ W .
P
• Consider the element (IdV ∗ ⊗ T )( ei ⊗ ei ).
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Trace Let T ∈ L(V ) be an operator, and consider it as a tensor in V ∗ ⊗V .
We can now obtain a number by contracting the upper and lower index
of such tensor. Show that this produces the trace of V . Notice that
since the contraction of tensors along factors does not depend on the
choice of basis, it follows that the trace of a matrix doesn’t change
when you conjugate the matrix by any invertible matrix.
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