general_linear_group

introduction

%%visits: 2 These are all the invertible matrices, all the matrices with non-zero determinants ## intuition It is a group.

rigour

GL(n, ℂ) = {A ∈ M_n(ℂ|det (A) ≠ 0}

Theorem. GL(n, ℂ) → ℂ^× is surjective. GL(n, ℝ) → ℝ^× is surjective.

The [[centre_of_a_group|centre]] of the general_linear_group is isomorphic to ℂ^× - Proof: See the lecture notes, but suffice to say, that the centre is only scaled identity matrices. We chose intereting Bs so that the commutative element AB = BA is singled out. See the lecture notes. Outline of the proof, the centre [[centre_of_a_group|centre]] is only diagonal matrices. ## exam clinic ## examples and non-examples ## resources tags :math:

backlinks

• index_of_files.pd