# How the original data can be written in the space defined by these M principal components?

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?

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.