"But... what if you can't do 'X' ? What if v, (0,0) and Av are not colinear?"
That's the point of the simulation -- to wiggle things and see what happens. You drag stuff in and out of colinearity on the third one down and see how repeated steps get pulled along the eigenspace, getting your steady state (I'd love to see an example using these sims to explain underdamped/overdamped systems).
I think the meat isn't in the words, but in the play.
That's the point of the simulation -- to wiggle things and see what happens.
Unfortunately, playing with the inputs to the simulation does not reveal to me any simple mental model of what's happening. About all I can figure out from the simulation is that the X- and Y- coordinates of each input and output are correlated (if I move a2 to the right, Av will move to the right, and so on). The relationship between A1, A2 and S1, S2 are not clear to me.
I have to agree with the parent that it doesn't explain much. It only states things and allows you to visualize the mathematical consequences. The visualization however does not assist me in understanding the model. I'm limited to understanding it from the pure mathematics.
A core part of explanation is allowing people to tie new knowledge to existing knowledge. Visualizations can be helpful because they allow people to take advantage of the inherent human visual system. For example, you might demonstrate how matrices can be used to transform object coordinates in a 3D system by displaying a 3D cube and allowing people to fiddle with the parameters. You could provide individual controls for translation and pitch/yaw/roll and show how those feed into matrix cells. By dragging one parameter, a person would see the cube begin to rotate, for example. That's an example of the kind of intuitive explanation that this page is missing; I can't connect the visualizations to any preexisting mental model of what should be happening.
True, but it doesn't supply the analytical tools so that you can find the correct A to produce an eigenvector without iteration to reduce error. Having said that you can reduce the linear system by hand if you are reasonably au fait with matricies etc and derive it yourself; but I suspect this "tutorial" is probably most beneficial for those who are struggling, i.e. the very people who wouldn't be able to leap to a solution on their own.
Comments
"But... what if you can't do 'X' ? What if v, (0,0) and Av are not colinear?"
That's the point of the simulation -- to wiggle things and see what happens. You drag stuff in and out of colinearity on the third one down and see how repeated steps get pulled along the eigenspace, getting your steady state (I'd love to see an example using these sims to explain underdamped/overdamped systems).
I think the meat isn't in the words, but in the play.
Unfortunately, playing with the inputs to the simulation does not reveal to me any simple mental model of what's happening. About all I can figure out from the simulation is that the X- and Y- coordinates of each input and output are correlated (if I move a2 to the right, Av will move to the right, and so on). The relationship between A1, A2 and S1, S2 are not clear to me.
I have to agree with the parent that it doesn't explain much. It only states things and allows you to visualize the mathematical consequences. The visualization however does not assist me in understanding the model. I'm limited to understanding it from the pure mathematics.
A core part of explanation is allowing people to tie new knowledge to existing knowledge. Visualizations can be helpful because they allow people to take advantage of the inherent human visual system. For example, you might demonstrate how matrices can be used to transform object coordinates in a 3D system by displaying a 3D cube and allowing people to fiddle with the parameters. You could provide individual controls for translation and pitch/yaw/roll and show how those feed into matrix cells. By dragging one parameter, a person would see the cube begin to rotate, for example. That's an example of the kind of intuitive explanation that this page is missing; I can't connect the visualizations to any preexisting mental model of what should be happening.
True, but it doesn't supply the analytical tools so that you can find the correct A to produce an eigenvector without iteration to reduce error. Having said that you can reduce the linear system by hand if you are reasonably au fait with matricies etc and derive it yourself; but I suspect this "tutorial" is probably most beneficial for those who are struggling, i.e. the very people who wouldn't be able to leap to a solution on their own.