1. Vector Space

What properties of vector space fail to hold the following:
The set of all ordered triples of real numbers with the operations:
U = (a,b,c), V = (x,y,z), r is a real number.
Vector Addition: U + V = (a+x, b+y, c+z)
Scalar Multiplication: r * U = (a, 1, c)

I am fine with addition properties. They hold true. I am confused with scalar multiplication.

Thanks.

2. Originally Posted by Kandiah
What properties of vector space fail to hold the following:
The set of all ordered triples of real numbers with the operations:
U = (a,b,c), V = (x,y,z), r is a real number.
Vector Addition: U + V = (a+x, b+y, c+z)
Scalar Multiplication: r * U = (a, 1, c)

I am fine with addition properties. They hold true. I am confused with scalar multiplication.

Thanks.
The scalar multiplication is not well defined Notice that

$2\vec{u}=[a,1,c]$ but

$2\vec{u}=\vec{u}+\vec{u}=[a+a,b+b,c+c]=[2a,2b,2c]$

3. Thanks for the response. Perhaps, since the scalar multiplication is not well defined I interpret differently and I may need further understanding. The scalar multiplication property is as follows: "If U (Vector) is in V (Vector Space) and c is a real number, then c*U is in V." Therefore, if a, b, & c in U are any real numbers and r is any real number, my interpretation is that r*U is not always equal to (a, 1, c). How do I interpret that r*U = (a, 1, C)?

4. Originally Posted by Kandiah
Thanks for the response. Perhaps, since the scalar multiplication is not well defined I interpret differently and I may need further understanding. The scalar multiplication property is as follows: "If U (Vector) is in V (Vector Space) and c is a real number, then c*U is in V." Therefore, if a, b, & c in U are any real numbers and r is any real number, my interpretation is that r*U is not always equal to (a, 1, c). How do I interpret that r*U = (a, 1, C)?
There is not good way to interpret it. In the original post you ask what vector space axioms fail. Well scalar multiplication fails to distibute over additon.

Vector space - Wikipedia, the free encyclopedia

Here is a list of axioms. It is the 6th one in the list.

5. Another property that fails to hold here: In any vector space $0\vec{v}= \vec{0}$. That is, the scalar, 0, times any vector is the 0 vector. By this definition of scalar multiplication, $0(a, b, c)= (a, 1, c)\ne (0, 0, 0)$ and (0, 0, 0) clearly is the additive identity.