Projection matrix: Difference between revisions

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==Intuition==

[[File:Projection of a vector onto the column space of a matrix.svg|thumb|A matrix, <math>\mathbf{A}</math> has its column space depicted as the green line. The projection of some vector <math>\mathbf{b}</math> onto the column space of <math>\mathbf{A}</math> is the vector <math>\mathbf{x}</math>]]

From the figure, it is clear that the closest point from the vector <math>\mathbf{b}</math> onto the column space of <math>\mathbf{A}</math>, is <math>\mathbf{Ax}</math>, and is one where we can draw a line orthogonal to the column space of <math>\mathbf{A}</math>. A vector that is orthogonal to the column space of a matrix is in the nullspace of the matrix transpose, so