Visualizing, identity matrix, scalar matrix, reflection matrix, diagonal matrix, zero matrix, shear matrix, orthogonal matrix, projection matrix, inverse of a matrix.
Chapters:
0:00 Visualize Matrix, but how ?
3:07 Identity Matrix
4:02 Scalar Matrix
6:23 Matrix in 3D
7:01 off-one Matrix
8:45 Reflection Matrix
10:54 Diagonal Matrix
14:00 Zero Matrix
Hopefully providing more intuition about matrix transformation on vectors and making the very abstract object of matrix, more relatable to us. This visual approach to matrix transformation also is the foundation to the Grand Finale of visualizing SVD.
This video wouldn’t be possible without the inspiration of the legendary 3b1b :
/ 3blue1brown
and the animation software - Manim, which he wrote:
/ 3blue1brown
and the Manim Community:
docs.manim.community/en/stabl...
Video Sins:
1. 1:16 I said “ we should think what happens to all vectors, as each everyone of them gets multiplied by the matrix “. I think my statement is factually incorrect, you can understand everything about a matrix if you know how it acts on the basis, i,j,k vectors. See the www.youtube.com/watch?v=kYB8I.... It’s a personal decision to put more vectors/arrows/dots/ on screens, because I think it increases our cognition to interpret the matrix, such as shearing, scaling, rotation transformation just becomes very visible. I also think this facilitates a very smooth transition to matrix transformation on pictures.
2. Sorry for calling identity matrix ‘boring’, it’s perhaps the most special matrix that there is. Consider every other matrices that exist does some transformation on vector, the identify matrix is the only one that does nothing. Many proofs in Lin Alg actually relies on assistances of identity matrix.
3. 5:40, I said “shape is just infinite number of dots sitting very closely together”. Sorry for using the word infinite so lightly, ... you get the idea. If I were to be more formal, I should have said a shape is collection of vertices sitting closely together: math.hws.edu/graphicsbook/c3/... .
4. Whenever it said “scale x axis by 3” on screen, it’s the lazy way of saying stretching vectors longer by factor of 3 purely in the x axis direction (horizontal direction).
Image Credits:
super mario, nyan cat, crypto punk, plants vs zombie-peashooter, and Obito