matrices as linear transformations
At first sight a matrix looks like nothing more than numbers arranged in a box. So people try to memorize it, and that is exactly why it stays confusing for years. A matrix is not a table; it is an action. More precisely, it is a function that moves every point of the plane at once: it stretches space, rotates it, and pushes it sideways. Once this picture settles in, it becomes much easier to see why matrix multiplication is defined the way it is, and what the eigenvalues later in the course are looking at.
Start by getting a feel for it. A grid covers the stage; change the four numbers of the matrix with the four sliders below and the whole grid bends and stretches together. Not just one point but the entire plane moves as a unit. That is what a transformation is, and a matrix is an order to change space in a particular way. The green arrow î and the orange arrow ĵ move with the grid. The important thing is that the grid never crumples at random. Straight lines stay straight after the transformation, parallel lines stay parallel, and the spacing stays even. Changing space while keeping it this orderly is what a matrix transformation does. After playing with the sliders, press "Reset" to return to the transformation that changes nothing.
So how can a transformation this complicated be written down with just a few numbers? The secret is surprisingly simple: follow only two arrows. Know where î, one unit to the right, and ĵ, one unit up, end up after the transformation, and you are done. Better still, those landing spots are exactly the two columns of the matrix: the first column is where î went, and the second is where ĵ went. Press "Land them" and the two arrows leave their original positions, which remain as dashed lines, and move to their new spots, each tip labeled with its coordinates. Compare those coordinates with the matrix below: the green coordinates match the first column and the orange ones match the second. The four numbers that looked meaningless were really coordinates saying "send î here and ĵ there." The î and ĵ buttons highlight one arrow at a time.
What good is knowing only î and ĵ? This is the heart of it. You do not need to know any other point separately. Any vector in the plane is ultimately some number of î's plus some number of ĵ's. The point (3, 2), for example, is 3 of î plus 2 of ĵ. A transformation preserves that mix, so the transformed (3, 2) is 3 of wherever î went plus 2 of wherever ĵ went. Drag the blue dot: on the transformed grid, the dashed green line lays out copies of the new î and the dashed orange line adds copies of the new ĵ to reach the point, and the equation below shows the counts. This is the linearity of the previous lesson. Fix the fate of î and ĵ and the infinitely many other points follow on their own, which is why the four numbers of a matrix can steer the whole space.
Suppose you have done one transformation and want to do another on top of it, say rotate first and then shear. Multiply the two matrices. Matrix multiplication simply means doing transformations one after another. The odd rule you learned in school for multiplying rows by columns computes the result of applying the transformations in sequence. That is why order matters. The figure deals with rotating then shearing and with shearing then rotating; press "Swap order" and the shape of the grid changes, and the two product matrices in the boxes below differ. Swap the order and the result changes. AB being different from BA was never a rule to memorize; it was obvious all along.
Now take a tour of a few standard transformations: rotation, scaling, shear (a sideways push), and reflection. Press the buttons one at a time and the grid changes accordingly, showing that each action corresponds to a single matrix. Every familiar action that changes space is, in the end, a matrix. Finally, press "Collapse" and space is squashed onto a single line. This is a special case: the value called the determinant, shown below, becomes 0. It signals that the transformation has crushed space flat. What this flattening means is the subject of the very next lesson, on the determinant.