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 $1(x_1, x_2) = (x_1, 0)$.
Best Answer
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.