algebra is a vector space 
over a field 
with a multiplication. The multiplication must be distributive and, for every 
and 
must satisfy
f(xy)=(fx)y=x(fy).
Note- by multiplication defined on teh evector space, we mean multiplication of two vectors to yield a number (or a vector maybe, depending on the case).
Examples of algebras include the algebra of real numbers, vectors and matrices, tensors, complex numbers, and quaternions.
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