how do you show that M_n(R) and R^(n^2) are homeomorphic? i defined a function f:M_n(R) -> R^(n^2) such that if you input any real matrix, you get out a vector that is set up as (a11, a12, ... a21, a22, ..., a_nn) so basically i go along the first row then the second row then 3rd row and so on of the matrix and put the elements in the order i get to them while doing this process. so i can see that this function is a bijection and now i need to show that both the function f and f^-1 are continuous and i am a bit lost on how to do that.
given any and we see that (in fact they're equal) since .
Isometries are also always injective since if then and thus . Thus, if is a surjective isometry it
must be bijective. But, we can also see that is an isometry. Indeed, if then by surjectivity we know that for some
and thus from where the conclusion follows.