Page 416 - Advanced Linear Algebra
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400 Advanced Linear Algebra
ker² ³ 0 º²c ³ ²!³ c ! ! Á 8 : »
and since ²²c ³ ²!³ c !³~ , it follows that ker ² ³~0 .
We now have quotient-space characterizations of the symmetric and
antisymmetric tensor spaces that do not place any restriction on the
characteristic of the base field.
Theorem 14.17 Let be a finite-dimensional vector space over a field .
=
-
)
1 The surjective linear map ¢; ²= ³ ¦ - ´ Á ÃÁ µ defined by
4 n Ä n 5 ~ v Ä v
ÁÃÁ ÁÃÁ
has kernel
0~ º ²!³ c ! ! Á 8 : »
and so
;²= ³
- ´ ÁÃÁ µ
0
The vector space ;²= ³°0 is also referred to as the symmetric tensor
space of degree of .
=
)
c
2 The surjective linear map ¢; ²= ³ ¦ - ´ Á ÃÁ µ defined by
4 ÁÃÁ n Ä n 5 ~ ÁÃÁ w Ä w
has kernel
0 ~ º²c ³ ²!³ c ! ! Á 8 : »
and so
;²= ³ c
- ´ ÁÃÁ µ
0
The vector space ;²= ³°0 is also referred to as the antisymmetric tensor
space or exterior product space of degree of .
=
The isomorphic exterior spaces (; ²= ³ and ; ²= ³°0 are usually denoted by
= and the isomorphic exterior algebras (;²= ³ and ;²= ³°0 are usually
denoted by = .
Theorem 14.18 Let be a vector space of dimension .
=
