(or How I Learned to Stop Worrying and Love the Matrix, part I)
For those of you charged with teaching a linear algebra course or still struggling to picture the complicated dance of mass arithmetic known as matrix multiplication, here’s a neat visualization tip:
Draw your matrices like this:
Once you’ve drawn them it should be immediately obvious how each entry in the new matrix gets filled: take the dot product of its row and column.
You immediately can visualize what size the new matrix will be:
And whether multiplication will be impossible because the sizes are incompatible:
Once you’ve multiplied matrices this way by hand a few times, it will start to feel natural to picture them this way. Then the next time you encounter matrix multiplication in an application or proof, even if no one’s asking you to compute the answer, you can lay matrices out this way in your mind’s eye to picture what exactly the multiplication does.