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## Linearity of the Inner Product

Any function of a vector (which we may call an

operatoron ) is said to belinearif for all and , and for all scalars and in , we have

A linear operator thus “commutes with mixing.”Linearity consists of two component properties,

additivity: , andhomogeneity: .The inner product is

linearin its first argument, i.e.

This is easy to show from the definition:The inner product is also

additivein its second argument, i.e.,

but it is onlyconjugate homogeneousin its second argument, sinceThe inner product

isstrictly linear in its second argument with respect torealscalars:Since the inner product is linear in both of its arguments for real scalars, it is often called a

bilinear operatorin that context.