thing is that if we had a linear map \$$\phi : \mathbb{R}^3 \rightarrow \mathbb{R}^2\$$ we would represent that as a matrix with 2 rows and 3 columns.

so by analogy the feasibility relation \$$\Phi : X \nrightarrow Y\$$ should be represented by a matrix with \$$|Y|\$$ rows and \$$|X|\$$ columns.

(fwiw I always imagine matrices as taking inputs through the "top side" and spitting out results through the "left side" – but I'm sure everyone has their own way of visualising this...)