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: a*b=overline{ab}. This is a commutative, non-associative division algebra of dimension 2 over the reals, and has no unit element.
Proof that mathbb{C},*) is a division algebra
For a proof that mathbb{R} 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 mathbb{R} and all vectors x, y, and z (also in mathbb{C}).
For distributivity:
:x*(y+z)=overline{x(y+z)}=overline{xy+xz}=overline{xy}+overline{xz}=x*y+x*z.
(similarly for right distributivity); and for the third and fourth requirements:ax)*y=overline{(ax)y}=overline{a(xy)}=overline{a}cdotoverline{xy}=overline{a}(x*y).
Non associativity of mathbb{C},*)
*:a * (b * c) = a * overline{b c} = overline{a overline{b c = overline{a} b c
*:a * b) * c = overline{a b} * c = overline{overline{a b} c} = a b overline{c}
So, if "a", "b", and "c" are all non-zero, and if "a" and "c" do not differ by a real multiple, a * (b * c)
eq (a * b) * c.
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