So the question is that if $X$ is in $M_2(\Bbb R)$ and has columns $u$ and $v$ so that $X = (u \ \ v)$, what would be [X^TX]_B in terms of dot product of u and u, v and v, and u and v. I feel like the basis would be {[1 0 /0 0], [0 0 /0 1], [0 1 /1 0]} but I don't really get how can I express them in terms of those products of u and v. Did I get the basis wrong or something? I supposed u as [u_1, u_2] and v as [v_1, v_2] and that was what I got from it. But this method was not really helpful with defining the basis in terms of u and v for me at least.
I don't know why but apparently I cannot write mathematical terms appropriately with the tablet so X^T is transpose of X, slash inside of the square brackets are meant to tell that ones before it and after it are in separate rows.
Thanks for all your help and time!! I really appreciate it cause this has been bothering me a lot!!
There's no need for bases. The quickest way to see what's going on is to use block-matrix multiplication: $$ X^TX = \pmatrix{u&v}^T \pmatrix{u&v} = \pmatrix{u^T\\v^T}\pmatrix{u&v} = \pmatrix{u^Tu&u^Tv\\v^Tu & v^Tv}. $$