# 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.

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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