This entire semester I still haven't been able to understand when to write a vector as a row or a column in a matrix depending on the situation. If any of you have any sort of insight please let me know.
For example:
Suppose $T: R^2 \rightarrow R^2$ is a linear map where
$T([1,1]) = [3,2] = b_1$
$T([2,3]) = [7,7] = b_2$
$T([x,y])=$ ?
Over here when I would be augmenting the two basis vectors into a matrix I would make it
$[b1,b2]$ (as column vectors) but I saw my prof turned them into row vectors.
The intuition that made me want to turn them into column vectors is because of the sense that each column is essentially the where $i$ and $j$ would go to after the first matrix multiplication (the intuition taught by 3blue1brown's video on matrix multiplication). Can someone help correct my wrong reasoning/logic?
First you have to write $[x,y]$ as a linear combination of the given vectors, $$[x,y]=\lambda[1,1]+\mu[2,3]\ .$$ Then the answer is $${\bf a}=\lambda[3,2]+\mu[7,7]\ .$$ These can be written in matrix form as $$[x,y]=[\lambda,\mu]\left[\matrix{1&1\cr2&3\cr}\right]\ ,\quad {\bf a}=[\lambda,\mu]\left[\matrix{3&2\cr7&7\cr}\right]$$ and hence $${\bf a}=[x,y]\left[\matrix{1&1\cr2&3\cr}\right]^{-1}\left[\matrix{3&2\cr7&7\cr}\right]\ .$$ It is probably more common to do all this with column vectors, but as you can see, row vectors work too. If you try it your way you should find that you get the same answer, but written as a column instead of a row.