I am studying vector spaces and I have problem understanding multiplication of a vector with a scalar from a field over which the vector space is defined.
It is given in a book which I refer that, for every $v$ of a vector space $V$, and $x$ of a field, there is $xv$ in the set $V$.
does it mean number of vectors in a vector space equal the number of elements in a field?
and there is an addition satisfying isomorphism?
I think just as the field elements are added, their maps in vectors get added following isomorphism under addition. Am i right?
I think addition of vectors and scalar multiplication work in tandem, but I cant find examples of finite vectors of finite dimensions, to see the addition and meaning of scalar multiplication.
Best Answer
If $V$ is a vector space over the field $\mathbf F$, then it must satisfy two properties, namely closure under addition and closure under multiplication.
For closure under multiplication, we demand that if $u \in V$, $a \in \mathbf F$, then $a \mathbf F \in V$. Note that the 'multiplication' needs to be defined beforehand.