My Math Forum scalar multiplication axioms

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 October 11th, 2011, 04:04 AM #1 Member   Joined: Jan 2010 Posts: 44 Thanks: 0 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.
 October 16th, 2011, 12:30 AM #2 Site Founder     Joined: Nov 2006 From: France Posts: 824 Thanks: 7 Re: scalar multiplication axioms The first and second axioms are not equivalent, but the two of them can be seen as equivalent to a jointed one: $S(\lambda x + y, z)= \lambda S(x,z) + S(y,z)$ I let you prove that this axiom can indeed substitute 1 + 2.

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