Example of a non-associative algebra
- Example of a non-associative algebra
This page presents and discusses an example of a non-associative division algebra over the real numbers.
The multiplication is defined by taking the complex conjugate of the usual multiplication: . This is a commutative, non-associative division algebra of dimension 2 over the reals, and has no unit element.
Proof that is a division algebra
For a proof that is a field, see real number. Then, the complex numbers themselves clearly form a vector space.
It remains to prove that the binary operation given above satisfies the requirements of a division algebra
* (x + y)z = x z + y z;
* x(y + z) = x y + x z;
* ("a" x)y = "a"(x y); and
* x("b" y) = "b"(x y);
for all scalars "a" and "b" in and all vectors x, y, and z (also in ).
For distributivity:
:.
(similarly for right distributivity); and for the third and fourth requirements:.
Non associativity of
*:
*:
So, if "a", "b", and "c" are all non-zero, and if "a" and "c" do not differ by a real multiple, .
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