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Suppose you apply PCA on the data $x_1,...,x_6$ and find that data can be fully described using M principal components $u_1,...,u_M$. How the original data can be written in the space defined by these M principal components?

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Assuming your basis vectors are sorted by decreasing eigenvalues, a sample $x_i$ can be transformed into the new space using dot products (projections), i.e. $[u_1^Tx_i,...,u_M^Tx_i]$. Note that, you first de-mean your data.

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