Chapter 4 General Vector Spaces Face Recognition ( á¹¢å ± 彐嬀)

Example 6Find the distance of the point x = (4, 1, 7) of R 3 from thesubspace W consisting of all vectors of the form (a, b, b).SolutionFrom previous Example 5:is an orthonormal basis for W.proj x W ( x u1) u1 ( x u2) u2 (4, 3, The distance from x to W isu 1 1, 0, 0), u(2x projWx (0, 4, 4) 0,3213)2,12LA 04_61(a6.4 Least-Squares Curves () Gauss 1797, Legendre 1806LA 04_62 Given a set of data points (x 1 , y 1 ), …, (x n , y n ), find a curvef (x) = a 0 + a 1 x + … + a m1 x m1 that minimizes the sum ofsquares of the errors (). Assume n m (explained later)AaT Min f ( a ) (a 0 + a 1 x 1 + … + a m1 xm11 y1 ) 2 + …+ (a 0 + a 1 x n + … + a m1 xm1n yn ) 2T a 11a 11 m 1mxnx na my 1nxnx1 1 na 1 m 1m 11 x x 0 y1 1 x x 0 y1ATy)AaT(AaaTATyy)(yAa— Given A and y, Unknown aaTTATyyTTy)(Aay)m1ynLA 04_63TAa 2 A y 0 for critical pointLeast-Square Curvesd: difference (a 0 + a 1 x 1 –y 1 ) 2 = (y 1 – a 0 – a 1 x 1 ) 2 LA 04_64Minimization of Least-Squares Curves by Calculus For simplified notation, assume 2 data points (x 1 , y 1 ), (x 2 , y 2 ),and a linear equation y = a 0 + a 1 xT T T T T Tf ( a ) a A Aa a A y y Aa y yT T T T a Ma a N N a y y M11M12 a0 N1 a0 T a0a1 a0a1 N1N2 y y M12M22 a1 N2 a1 22T M11a0 2 M12a0a1 M22a1 2 N1a0 2 N2a1 y yTT T T T T Twhere M A A , N A y , N ( A y ) y A , M M f ( a ) 2 M11a0 2 M12a1 2 N1 a0 M11M12 a0 N1 2 2( ) f a12 22 12 2 2 2 M M a NM12a0 M22a1 N2 a1 f ( a ) a 2 Ma 2 N 2 TA