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October 11th, 2011, 04:04 AM   #1
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scalar multiplication axioms

Scalar multiplication is defined generally as a function ,where , - linear space, for which 5 axioms are true:

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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October 16th, 2011, 12:30 AM   #2
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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:

I let you prove that this axiom can indeed substitute 1 + 2.
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