Given a matrix M such that its columns are the vectors of a new basis with respect to another basis B.
To find the coordinates of v in the other basis, we can simply take $M[v]_M = [v]_B$.
Let me give an example of M
I believe they are linearly independent(i just pulled out some random number off my head and tested), but the numbers aren't that important.
What i am confused about is we know that the columns of M form a set of basis vectors but when doing $M[v]_m$ matrix multiplication, we iterate within each $row_i$ of M for each value in the corresponding row of the output vector instead.
Now, i learn that, in my school's materials convention, we represent linear functionals as row vectors instead, since column vectors are for things like coordinate vectors and this makes sense to me at least here, but above, i am using a basis matrix's rows like linear functionals?
So yeah, is it just "it is how it is because matrix multiplication rules", or is there some special property or something about rows in matrices.