A one-to-one map between $M_{n\times n}$ and the $\mathbb{R}^{n^{2}}$?

Question

I've been trying to think in something that can make this happen, but i'm not get anywhere. Plus, i have to show something through this map that can make this space($M_{n\times n}$) a metric space, so a think to use the trace of a matrix. Is that correct?

Answer 1

The trace of matrix is invalid since it reduces the dimension from $n^2$ to $n$. You can define the set of $n\times n$ matrices with Frobenius norm and reorder the rows of the matrices back to back and define the Euclidean $2$-norm on $\Bbb R^{n^2}$. In this manner the two metric spaces would become homeomorphic.

Answer 2

Just take $$ f:A\to (a_{1,1},a_{1,2},\dots,a_{1,n},a_{2,1},\dots a_{2,n},a_{3,1}\dots,a_{n,1},a_{n,2},\dots, a_{n,n}) $$ i.e just rearrange the entries of $A$ into a vector in $\mathbb{R}^{n^2}$. Then define $d(A,B):=||f(A)-f(B)||$.