Lecture 17 Orthogonal basis Orthogonal matrix Gram Schmidt Orthonormal vectors Matrix Q has orthonormal columns Example Project onto its column space Properties of a projection matrix Normal equation Gram Schmidt A QR If the matrix is square then the column space is the whole space and the projection matrix onto the whole space is the identity matrix Projection matrices are symmetric Multiplying a projection matrix by itself changes nothing Start with independent vectors a and b Our goal is to produce orthogonal vectors A and B and then orthonormal vectors A A and B B Now we need to make B orthogonal to A B is equal to the error vector B b Example The column space of A is the same as the column space of Q If there is a third vector c we need to nd the orthonormal vector C C The formula for C is
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