- #1
kidsasd987
- 143
- 4
I have a question about weights of a basis set with respect to the other basis set of one specific vector space.
It seems the weights do not covert linearly when basis sets convert linearly. I've got this question from the video on youtube "linear transformation"
Let's consider a vector space V spanned upon field K.
Then its element x∈V can be expressed in the form of a linear combination with a given basis set and its corresponding weights.
x=ΣaixiNow if we were to express a vector x with respect to the new basis x_hati,
we would express the same vector as
x=Σbix_hatiWe can derive the relation
Λ(xi)=x_hati
bi=Λ-1aiI did this to cylindrical coordinate, and because
https://wikimedia.org/api/rest_v1/media/math/render/svg/cf553bbb290f2b6ad76c9cce12f8807d43ab09ee
and according to the equation from video
bi=Λ-1ai
inverted matrix (transpose of matrix above applied to <x,y,z> should give converted weight bi but this isn't the case.
rho=sqrt(x^2+y^2) and phi=tan^-1(y/x) and they are not linear functions.could anyone tell me why is it so?
It seems the weights do not covert linearly when basis sets convert linearly. I've got this question from the video on youtube "linear transformation"
Let's consider a vector space V spanned upon field K.
Then its element x∈V can be expressed in the form of a linear combination with a given basis set and its corresponding weights.
x=ΣaixiNow if we were to express a vector x with respect to the new basis x_hati,
we would express the same vector as
x=Σbix_hatiWe can derive the relation
Λ(xi)=x_hati
bi=Λ-1aiI did this to cylindrical coordinate, and because
https://wikimedia.org/api/rest_v1/media/math/render/svg/cf553bbb290f2b6ad76c9cce12f8807d43ab09ee
and according to the equation from video
bi=Λ-1ai
inverted matrix (transpose of matrix above applied to <x,y,z> should give converted weight bi but this isn't the case.
rho=sqrt(x^2+y^2) and phi=tan^-1(y/x) and they are not linear functions.could anyone tell me why is it so?