Linear Algebra 21d: Rotation Matrices in a Non-Cartesian Basis

[stuffenjoyer]

Excellent Video. Honestly, all of them are excellent. Thanks.

[tangolasher]

+MathTheBeautiful Hi, I'm completely bewildered by why you multiply the matrix elements by 2 and 1/2 *diagonally.* The rotation matrix columns are just the two basis vectors e1 and e2, after the rotation, right? And you scaled the first basis vector e1 by 2. So shouldn't you just multiply the first column of the rotation matrix by 2? Or maybe divide the second by 1/2? Even multiplying a row by some number would make sense to me, but why on earth are you multiplying different rows of each column *diagonally?* So utterly confused, sorry. Any help would be much appreciated.

[mrmedmor]

first thank you for all your BEAUTITUL MATH videos
i am just confused about the decomposition of Re1, i see it as 2cos(alpha) 2sin(alpha)
and on the bord it seems to me that the components of Re1 are twice bigger than those for Re2, but on the matrix we put cos(alpha) for both

[matthewsarsam]

This new matrix isn’t unitary, but still represents the same exact transformation. So if I take a vector V in the old basis and use its components with the old matrix, and the same vector V in this new basis, the output will have different components but represent the same vector. So therefore length is preserved with respect to the new basis, but the matrix isn’t unitary. I guess the question is, if the matrix isn’t unitary, then it can’t preserve length with respect to an orthonormal basis. Please see if I’m on the right path here thank you