2014-38.md

course	course_year	question_number	tags	title	year
Linear Algebra	IB	38	IB 2014 Linear Algebra	Paper 1, Section I, G	2014

State and prove the Steinitz Exchange Lemma. Use it to prove that, in a finitedimensional vector space: any two bases have the same size, and every linearly independent set extends to a basis.

Let $e_{1}, \ldots, e_{n}$ be the standard basis for $\mathbb{R}^{n}$. Is $e_{1}+e_{2}, e_{2}+e_{3}, e_{3}+e_{1}$ a basis for $\mathbb{R}^{3} ?$ Is $e_{1}+e_{2}, e_{2}+e_{3}, e_{3}+e_{4}, e_{4}+e_{1}$ a basis for $\mathbb{R}^{4} ?$ Justify your answers.