Axiom 10 of Vector Spaces

In summary, Axiom 10 of Vector Spaces, also known as the axiom of scalar multiplication, states that the product of any scalar and vector will result in another vector in the same vector space. This implies that scalar multiplication in a vector space is closed, meaning that it is a well-defined operation. It is an important property in the study of vector spaces as it allows for the application of mathematical concepts and techniques. Axiom 10 can also be generalized to higher dimensions and holds true for all vector spaces without any exceptions.
  • #1
elements
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So I understand how to prove most of the axioms of a vector space except for axiom 10, I just do not understand how any set could fail the Scalar Identity axiom; Could anybody clarify how exactly a set could fail this as from what I know that anything times one results in itself

1u = u
1(x,y,z)=(x,y,z)
1(1,2,3) = (1,2,3)
1 (1,0,...,1) = (1,0,...,1)

I don't see how you could ever end up in a situation where you could end up with

1VV
 
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  • #2
elements said:
So I understand how to prove most of the axioms of a vector space except for axiom 10, I just do not understand how any set could fail the Scalar Identity axiom; Could anybody clarify how exactly a set could fail this as from what I know that anything times one results in itself

1u = u
1(x,y,z)=(x,y,z)
1(1,2,3) = (1,2,3)
1 (1,0,...,1) = (1,0,...,1)

I don't see how you could ever end up in a situation where you could end up with

1VV
It can happen if you have a vector space with an unusual kind of scalar multiplication. Keep in mind that a vector space consists of a set of vectors over some field (often, the field of real numbers ##\mathbb{R}## or the field of complex numbers ##\mathbb{C}##), together with operations for vector addition and for multiplication by a scalar.

If scalar multiplication is defined like this
##k \cdot <x, y> = <kx, 0>##
then there is no scalar k for which ##k \cdot <x, y> = <x, y>##.
 

What is Axiom 10 of Vector Spaces?

Axiom 10 of Vector Spaces, also known as the axiom of scalar multiplication, states that for any scalar c and any vector v, their product cv is also a vector in the vector space.

What does Axiom 10 imply about scalar multiplication in a vector space?

Axiom 10 implies that scalar multiplication in a vector space is closed, meaning that the product of any scalar and vector will always result in another vector in the same vector space.

Why is Axiom 10 important in the study of vector spaces?

Axiom 10 is important because it is one of the fundamental properties that defines a vector space. It ensures that scalar multiplication is a well-defined operation in a vector space, allowing for the application of various mathematical concepts and techniques.

Can Axiom 10 be generalized to higher dimensions?

Yes, Axiom 10 can be generalized to higher dimensions. It applies to vector spaces of any dimension, as long as the scalar and vector involved are in the same vector space.

Are there any exceptions to Axiom 10?

No, Axiom 10 holds true for all vector spaces. It is a universally accepted property that defines the nature of vector spaces.

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