Global analysis. Exercises 5
1) Let V be a vector space of dimension n, A  2
V , B  V . Find the components
of the tensor A  B.
2)Let V and W be vector spaces of dimensions n and m, and with the bases e1, ..., en
and f1, ..., fm. Let A : V  V and B : W  W be linear maps. Define the linear
map A  B : V  W  V  W by A  B(v  w) = A(v)  B(w). Prove that the
matrix of A  B in the basis e1  f1, e1  f2, ..., e1  fm, e2  f1, e2  f2, ..., e2 
fm, ..., en  f1, en  f2, ..., en  fm of V  W has the form






a11B a12B . . . a1nB
a21B a22B . . . a2nB
. . . . . . . . . . . .
an1B an2B . . . annB






,
where A = (aij) and B = (bij) are the matrices of the linear maps A and B.
3) Prove that V  R V .
4) Let V be a vector space, T a tensor of type (1, 0) on V and S a tensor of type
(0, 1) on V . What gives the contraction of the tensor product T and S.
5) Let A  2
R2
 (R2
)
be the tensor with the components:
A11
1 = 3, A11
2 = 0, A12
1 = 2, A12
2 = 1,
A21
1 = 0, A21
2 = 1, A22
1 = 0, A11
1 = 5.
Find the contractions of the first and the second upper indices with the down index.
6) Let V be a vector space, A  r
V s
V 
. Let e1, ..., en and e1 , ..., en be bases
of V and let d1
, ..., dn
and d1
, ..., dn
be the dual bases. Find the relation between
the components Ai1...ir
j1,...,js
and A
i1...ir
j1,...,js
of the tensor A in these bases.
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