Question

What is the highest possible dimension of a subspace of M_n (R) (set of n×n matrices with real coefficients with its usual vector space structure) that only contains invertible matrices (and 0) ?

Answer 1

the general linear group over set of real numbers  R  is the group of n×n invertible matrices of real numbers, and is denoted by GL_n(R) or GL(n, R).So here we are asked to collect all invertible matrix that means all matrices with determinant zero.

You should notice that the determinant is a continuous mapf:M(n,R)≡Rn

f:M(n,R)≡Rn2→R,

Let us define map f(X) =det(X)  

Note that GL(n, R)= f^-1(R-{0})

And we know from real analysis that inverse image of open set under a continous map is open So from here it can be easily said that the pre-image by continuous map of open set is an open set then GL(n, R) will be an open subet of R^n2 this gives us idea that dimension(Gl(n, R) =n^2.

So dimension of desired subspace is n^2.

GL(n,R) is an open set.