Question

Question: Let be a none zero real number. Prove that {1, v} is linearly independent over Q if and only if v is irrational.

Short answer

Answer

Thus, $\left\{1,v\right\}$is linearly independent over Q if and only if v is irrational.

Step-by-step solution

Explanation

Given that v is a non-zero real number.

If part,

Let {1,v} is linearly independent.

We have to show that is irrational.

Only if part.,

Let v is irrational. we have to show that is {1,v} linearly independent.

Solution

$v\ne 0$Here we solve if part,

Suppose v is rational.

Since.

$\therefore \frac{-1}{v}\left(\ne 0\right)\in Q$

Consider

$1.1+\left(-\frac{1}{v}\right)v=0$

But$1\ne 0,$

$\frac{-1}{x}\ne 0$

Which is a contradiction as is linearly independent

Hence, v is irrational.

Now here we solve only if part,

Let$a·1+b·v=0,a,b\in Q$

Suppose $b\ne 0$

$$\begin{gathered} \therefore v=-\frac{a}{b} \\ \because a,b\in Q \\ \therefore -\frac{a}{b}\in Q \\ =v\in Q \end{gathered}$$

This is a contradiction, as we have assumed v to be irrational.

Hence,{1,v} is linearly independent.