307Notes-Chapter3 - Chapter III Orthogonality 65 III...

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Chapter III Orthogonality 65
III Orthogonality III.1 Orthogonality and Projections III.1.1 Orthogonal vectors Recall that the dot product, or inner product of two vectors x = 2 6 6 6 4 x 1 x 2 . . . x n 3 7 7 7 5 y = 2 6 6 6 4 y 1 y 2 . . . y n 3 7 7 7 5 is denoted by x · y or h x , y i and defined by x T y = x 1 x 2 · · · x n 2 6 6 6 4 y 1 y 2 . . . y n 3 7 7 7 5 = n X i =1 x i y i Some important properties of the inner product are symmetry x · y = y · x and linearity ( c 1 x 1 + c 2 x 2 ) · y = c 1 x 1 · y + c 2 x 2 · y . The norm, or length, of a vector is given by k x k = p x · x = v u u t n X i =1 x 2 i An important property of the norm is that k x k = 0 implies that x = 0 . The geometrical meaning of the inner product is given by x · y = k x kk y k cos( ) where is the angle between the vectors. The angle can take values from 0 to . The Cauchy–Schwarz inequality states | x · y |  k x kk y k . It follows from the previous formula because | cos( ) | 1. The only time that equality occurs in the Cauchy–Schwarz inequality, that is x · y = k x kk y k , is when cos( ) = ± 1 and is either 0 or . This means that the vectors are pointed in the same or in the opposite directions. 66
III.1 Orthogonality and Projections The vectors x and y are orthogonal if x · y = 0. Geometrically this means either that one of the vectors is zero or that they are at right angles. This follows from the formula above, since cos( ) = 0 implies = / 2. Another way to see that x · y = 0 means that vectors are orthogonal is from Pythagoras’ formula. If x and y are at right angles then k x k 2 + k y k 2 = k x + y k 2 . But k x + y k 2 = ( x + y ) · ( x + y ) = k x k 2 + k y k 2 + 2 x · y so Pythagoras’ formula holds exactly when x · y = 0. To compute the inner product of (column) vectors X and Y in MATLAB/Octave we use the formula x · y = x T y . Thus the inner product can be computed using X’*Y . (If X and Y are row vectors, the formula is X*Y’ .) The norm of a vector X is computed by norm(X) . In MATLAB/Octave inverse trig functions are computed with asin(), acos() etc. So the angle between column vectors X and Y could be computed as > acos(X’*Y/(norm(X)*norm(Y))) III.1.2 Orthogonal subspaces Two subspaces V and W are said to be orthogonal if every vector in V is orthogonal to every vector in V . In this case we write V ? W . In this figure V ? W and also S ? T . 67
III Orthogonality A related concept is the orthogonal complement. The orthogonal complement of V , denoted V ? is the subspace containing all vectors orthogonal to V . In the figure W = V ? but T 6 = S ? since T contains only some of the vectors orthogonal to S . If we take the orthogonal complement of V ? we get back the original space V : This is certainly plausible from the pictures. It is also obvious that V ( V ? ) ? , since any vector in V is perpendicular to vectors in V ?

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