In theory, PCA produces the same number of principal components as there are features in the training dataset. Principal components are orthogonal projections of data onto lower-dimensional space. But in very large datasets (where the number of dimensions can surpass 100 different variables), principal components remove noise by reducing a large number of features to just a couple of principal components. In the small 2-dimensional example above, we do not gain much by using PCA, since a feature vector of the form (feature1, feature2) will be very similar to a vector of the form (first principal component (PCA1), second principal component (PCA2)). Getting principal components is equivalent to a linear transformation of data from the feature1 x feature2 axis to a PCA1 x PCA2 axis. PCA allows us to go a step further and represent the data as linear combinations of principal components. The original data can be represented as feature vectors. The second component is orthogonal to the first, and it explains the greatest amount of variance left after the first principal component. The first principal component is computed so that it explains the greatest amount of variance in the original features. The principal components are vectors, but they are not chosen at random.
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