# Thread:

1. ## scalar multiplication axioms

Scalar multiplication is defined generally as a function $S:E \times E \longrightarrow K$,where $K\in(R,C)$, $E$ - linear space, for which 5 axioms are true:

$1. S(x+y,z)=S(x,z)+S(y,z)\\2. S(\lambda x,y)=\lambda S(x,y)\\3. S(x,y)=\bar {S(x,y)}\\4. S(x,x)\geq 0\\5. S(x,x)=0 \Rightarrow x=\bar{0}$

It is neccessary for any function that defines scalar multiplication, that all of these axioms are true.
I've noticed that some sources offer just 4 axioms - the first and the second is joined into one. Does it mean that the first and the second axiom is equivalent?
If not, then there must be a function for which the first axiom is false, but the remaining ones is true. But I cannot think of such an example.
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