W1D5 Tutorial1 Change of basis question

In the tutorial, the equation given is Y=XW. But why isn’t it X=WY?
The formula of coordinates transformation is x=Wy. If we concatenate a group of x’s and y’s together then it should be X=WY.

I’m assuming X, Y, W are all pretty arbitrary so it doesn’t really matter which one is before/after - transformation data. Where did you find x = Wy? I could not find it in the tutorial…

Hi, thanks for the reply! But still WY and YW would mean completely different things…
x=Wy is what I learned in linear algebra, it’s not in the tutorial.

Thanks for your reply!

It makes more sense to me when the vector of the new basis is on the right. In the tutorial, W is the new basis (orthogonal, if it is one of the eigenvectors of X). What was the case in your textbook…?

Maybe somebody else could give a better/clearer answer… Sorry!

Hi,

I don’t remember the exact context of this but perhaps the issue is that in the standard formulation, x = Wy, x and y are both column vectors. However, to form the matrices (X,Y) x and y are each transposed (so made into row vectors) and then concatenated vertically. So a transposed version of the earlier equation, X = Y W^T, applies here to the matrices (X,Y). Then since W is a co-ordinate transform its inverse is equal to its transpose so by post-multiplying by W we get Y = XW.

Hi, in the textbook it is just a general formula for change of basis…x=Wy or y=W^{-1}x where W is the matrix of basis transformation that is defined by

(e_1', e_2',...,e_n')=(e_1, e_2, ...,e_n)W

But I think Alex gives a pretty good explanation!

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Hi Alex, thanks for your explanations! I didn’t realize that in this case each row is a data vector…it seems that in linear algebra people tend to write horizontally concatenated column vectors…thanks!

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