der Universitat Munchen Set 3
Prof. Dr. B. Pareigis
Problem set for
Advanced Algebra
(9) (Tensors in physics: # 9,10,11) Let V be a nite dimensional vector space
over the eld K and let V
be its dual space. Let t be a tensor in V :::
V V
:::V
=V r
(V
) s
.
Show that for each basis B =(b
1
;:::;b
n
) and dual basis B
=(b 1
;:::;b n
)
there is a uniquely determined scheme (a family or an (r+s)-dimensional
matrix) of coeÆcients(a(B) i1;:::;ir
j1;:::;js
) with a(B) i1;:::;ir
j1;:::;js
2K such that
(1) t= n
X
i
1
=1 :::
n
X
i
r
=1 n
X
j
1
=1 :::
n
X
j
s
=1 a(B)
i1;:::;ir
j
1
;:::;js b
i
1
:::b
ir b
j1
:::b js
:(1)
(10) Showthatforeachchangeofbases L:B !C withc
j
= P
i
j b
i
(withinverse
matrix ( i
j
))the following transformation formula holds
(2) a(B)
i1;:::;ir
j
1
;:::;j
s
= n
X
k
1
=1 :::
n
X
kr=1 n
X
l
1
=1 :::
n
X
ls=1
i
1
k
1 :::
i
r
kr
l
1
j
1 :::
l
s
j
s a(C)
k1;:::;kr
l
1
;:::;ls
(11) Show that every family of schemes of coeÆcients
(a(B)jB basis of V)
with a(B) = (a(B) i
1
;:::;i
r
j
1
;:::;j
s
) and a(B) i
1
;:::;i
r
j
1
;:::;j
s
2 K satisfying the transformation
formula (2) denes a unique tensor (independent of the choice of the basis)
t2V r
(V
) s
such that (1)holds.
Rule for physicists: A tensor is a collection of schemes of coeÆcients that
transform according to the transformation formula for tensors.
(12) Show that (A;r : AA ! A; : K ! A) is a K-algebra if and only if A
with the multiplication AA
! A A r
! A and the unit (1) is a ring
and : K ! Cent(A) is a ring homomorphism into the center of A, where
Cent (A):=fa2Aj8b2A:ab=bag.
