Question

Question: If $\left\{v_1,v_2,v_{3,-------}{v}_n\right\}$is a basis of v, prove that $v_i\ne 0_v$for every i.

Short answer

Answer

It is proved that$v_i\ne 0_v$for every i.

Step-by-step solution

Explanation

Let$B=\left\{v_1,v_2,v_{3,-------}{v}_n\right\}$is a basis for v.

We have to show that,

$v_i\ne 0_v$( Zero of vector space V) ,$\forall i$

Concept

Since$B=\left\{v_1,v_2,v_{3,-------}{v}_n\right\}$is a basis for V.

$\therefore B$is linearly independent

i.e if$a_1{v}_1+a_2{v}_2+\_\_\_\_\_+a_n{v}_n=0$

then $a_i=0$_____(1)

Solution

We prove it by contradiction

Letat least on i such that$v_i=0$.

Now, we consider

$0·v_1+0·v_2+--+1·v_i+--+0·v_n=0$

Here,

$a_1=0,a_2=0,---,a_n=0$

But$a_i\ne 1\ne 0$

Which contradicts (1)

Hence, our assumption is wrong.

Hence,$v_i\ne 0_v\forall i$,