# Decide whether the specified vectors are a linearly independent system or not.

Decide whether the specified vectors are a linearly independent system or not. $$\frac{1}{3}, \quad \sqrt{2}$$ in the $$\mathbb{Q}$$ vector space $$\mathbb{R}$$.

I only need to show, that $$\lambda_1 \cdot \frac{1}{3} + \lambda_2 \cdot \sqrt{2} = 0$$
If $$\lambda_1 = \lambda_2 = 0$$ is the only solution, then it's linear independent, otherwise dependent, right?

• yes it is all you need to show. – Enkidu May 9 '19 at 8:44
• Note that also $\lambda_i\in \mathbb{Q}$. – denklo May 9 '19 at 8:45

Let $$\lambda_1 \cdot \frac{1}{3} + \lambda_2 \cdot \sqrt{2} = 0$$ and $$\lambda_1, \lambda_2 \in \mathbb Q$$. Assume that $$\lambda_2 \ne 0$$. Then we get $$\sqrt{2}=- \frac{\lambda_1}{3 \lambda_2}$$.
Since $$\frac{\lambda_1}{3 \lambda_2}$$ is rational, we have a contradiction. Thus $$\lambda_2 = 0$$. It follows that $$\lambda_1 = 0$$.
Conclusion: $$\frac{1}{3} , \sqrt{2}$$ are linearly independent in the $$\mathbb Q$$- vector space $$\mathbb R.$$