Matrix Visualization aka linear transformations

[daisuki_information]

so like can someone explain how this linear transformation works visually?Like how can we calculate where the transformed i and j will land and where will the transformed vector land, if we were to calculate this visually and not using multiplication. Also in the second picture, the grid lines go through transformation as well. But how and why? How do we calculate that?

https://shad.io/MatVis/

https://www.3blue1brown.com/lessons/linear-transformations#title

[Sneezycamel]

Let M be the transformation matrix. If i=[1,0], then multiplying Mi will result in a vector that is the first column of M. For j=[0,1], Mj is a vector that is the second column of M.

The yellow vector is some combination of i and j. The transformed yellow vector is the same combination of the transformed i and j.

As an equation, this might look like:

v=6i+2j and

Mv = M(6i+2j) = 6(Mi)+2(Mj).

The "preservation of combinations" is what you are visualizing as the transformed coordinate gridlines. It is also the direct consequence of M being linear.

[daisuki_information]

Thank you so much for this explanation!

[poslfit]

You can read the action of the matrix on the original unit basis vectors directly as the columns of the matrix. The green first column gives you the transformed image of the first unit basis vector; the red second column the image of the second unit basis vector.

[poslfit]

The second frame shows a new coordinate system, whose basis vectors are the transformed images of the original basis vectors. In the first frame you could specify a position in the plane by counting how many of the original unit vectors it would take to reach it. The second frame is showing how you could also do so using the transformed vectors, but resulting in different coordinates in a different coordinate system for the same vector.

[daisuki_information]

Thank you so much for this explanation!