Thread: 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.

Appreciate your help.
Thanks.

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.

Appreciate your help.
Thanks.

The scalar multiplication is not well defined Notice that

but

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)?

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.

Another property that fails to hold here: In any vector space . That is, the scalar, 0, times any vector is the 0 vector. By this definition of scalar multiplication, and (0, 0, 0) clearly is the additive identity.