# Proving a Set is not a Vector Space (Using a Nonstandard Definition of Scalar Multiplication)

I have a question about proving that a set is not a vector space. In my Linear Algebra textbook, we have this example:

"Let V be the set of all ordered pairs of real numbers, with the standard operation of addition and the nonstandard definition of scalar multiplication listed below."

$$c( x_1 , x_2) = ( cx_1, 0)$$

The book says that the first 9 axioms pass, but that the 10th axiom (Multiplicative Identity Property) fails because:

$$1(1,1) = (1,1) \neq (1,0)$$

But I am not certain as to why... Is it because $$1(x_1, x_2) = (1x_1,1x_2)$$ which simplifies to $$(x_1,x_2)$$, thus resulting in $$(1,1)$$ as opposed to $$(1,0)$$?

Though, I have doubts, because at the same time I can see 1 being c in the way scalar multiplication is defined, and thus $$1(x_1, x_2) = (x_1, 0)$$.

According to the axiom that we mentioned, you should have $$1(1,1)=(1,1)$$. But what we actually have for this scalar product is $$1(1,1)=(1,0)$$. Since $$(1,1)\neq(1,0)$$, that axiom doesn't hold and therefore we don't have a vector space.