Question

Prove that any non-trivial vector space has an infinite number of distinct elements.​

Answer 1

Step-by-step explanation:

If V is not the trivial vector space let v∈V, v≠0 . Then show that the vectors λv (λ∈R) are all distinct.

Answer 2

Given : a non-trivial vector space.

To show : non-trivial vector space contains infinitely many elements.

Step-by-step explanation:

Let V be a vector space and x be the non-zero elements in it.

Again consider the set,

$A=\{tv :t\in \mathbb R\}$   where $v\in V$

SInce V is closed under scalar multiplication so A is contained in V.

We have to show A contains all distinct elements.

Let $s,t\in \mathbb R$ such that,

$tx=sx\implies x(t-s)=0\implies s-t=0$  since $x\neq 0$

That is s=t, which shows that no two elements are equal in A.

Now since $\mathbb R$ is infinite therefore A contains infinitely many elements.

Hence proved that non-trivial vector space V has an infinite number of distinct elements.