State and prove the Steinitz Exchange Lemma.

Deduce that, for a subset $S$ of $\mathbb{R}^{n}$, any two of the following imply the third:

(i) $S$ is linearly independent

(ii) $S$ is spanning

(iii) $S$ has exactly $n$ elements

Let $e_{1}, e_{2}$ be a basis of $\mathbb{R}^{2}$. For which values of $\lambda$ do $\lambda e_{1}+e_{2}, e_{1}+\lambda e_{2}$ form a basis of $\mathbb{R}^{2} ?$