Let: $ u: \mathbb{R}^2 \rightarrow \mathbb{R}^3$ be defined by: $$ u(x,y)=(x+2y, 2x-y, 2x+ 3y)$$ Give the matrix $M[u]$ in the canonical base of its definition space.
This question might seem sort of stupid, but it was part of an exam we had a few days back, and we never had $u$ under this form to work with before. I thought of it this way: $u_1 = x+2y$; $u_2= 2x - y$; $u_3= 2x + 3y$;
And so I placed each $u_i$ in its respective column in the matrix (the coefficients that is). But according to the answer sheet that the professor posted, the coefficients of $x$ and $y$ are the ones placed column-wise. Any thoughts or explanations as to why they should be written like that?
Because what we need is to write $M(u)(x,y)^T=(x+2y,2x-y,2x+3y )^T$ and we need the matrix abd vector dimensions to agree, so $M(u)$ mist be a $3\times 2$ matrix, so we place the coefficients as the rows of $M$