Page 486 - Advanced Linear Algebra
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470 Advanced Linear Algebra
( ~ ) consider
% ~ c c c
11. Describe the left regular representation of the quaternions using the ordered
basis 8 ~² Á Á Á ³ .
be the group of permutations bijective functions of the ordered set
)
(
12. Let :
? ~ ²% ÁÃÁ% ³, under composition. Verify the following statements.
Each defines a linear isomorphism : on the vector space with
=
basis ? over a field - . This defines an algebra homomorphism
¢ -´: µ ¦ B - ²= ³ with the property that ² ³ ~ . What does the matrix
look like? Is the representation faithful?
representation of a :
13. Show that the center of the algebra B - ²= ³ is
A~ ¸ -¹
14. Show that B - ²= ³ is simple if and only if dim ²= ³ B .
15. Prove that for , the matrix algebras C ²-³ are central and simple.
16. An element ( is left-invertible if there is a ( for which ~ , in
which case is called a left inverse of . Similarly, ( is right-
invertible if there is a ( for which ~ , in which case is called a
right inverse of . Left and right inverses are called one-sided inverses
and an ordinary inverse is called a two-sided inverse . Let ( be
algebraic over .
-
a Prove that ~ for some £ if and only if ~ for some £ .
)
Does necessarily equal ?
)
b Prove that if has a one-sided inverse , then is a two-sided inverse.
Does this hold if is not algebraic? Hint : Consider the algebra
²-´%µ³.
( ~ B -
c Let Á ( be algebraic. Show that is invertible if and only if
)
and are invertible, in which case is also invertible.
