Homotopic maps in the general linear group

Prove that the maps \mathrm{GL}(n,\mathbb R)\times \mathrm{GL}(n,\mathbb R)\longrightarrow \mathrm{GL}(2n,\mathbb R) that send (A,B) to the block matrices \begin{pmatrix} A & 0 \\ 0 & B \end{pmatrix} and \begin{pmatrix} AB & 0\\ 0 & 1\end{pmatrix} are homotopic.

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Homotopic maps in the general linear group

Stable equivalence

Given a topological space X we can construct its suspension \Sigma X. This construction can be lifted to a functor from topological spaces to topological spaces that acts on arrows by sending a continuous map f : X \to Y to its suspension \Sigma f : \Sigma X \to \Sigma Y which is just f in each “level” of the suspensions.

Problem 1: Find two continuous non homotopic maps such that their suspensions are homotopy equivalent.

Problem 2: Find a non nullhomotopic map such that its suspension is nullhomotopic, but now we require the domain and the codomain to be connected!

Stable equivalence