I have a question about weights of a basis set with respect to the other basis set of one specific vector space.(adsbygoogle = window.adsbygoogle || []).push({});

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=Σa_{i}x_{i}

Now if we were to express a vector x with respect to the new basis x_hat_{i},

we would express the same vector as

x=Σb_{i}x_hat_{i}

We can derive the relation

Λ(x_{i})=x_hat_{i}

b_{i}=Λ^{-1}a_{i}

I 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

b_{i}=Λ^{-1}a_{i}

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?

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# I Linear transformation of a given coordinate

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