Principles of Mathematics in Operations Research
30 2 Preliminary Linear Algebra http://en.wiklbooks.org/wiki/Algebra/Linear_transformations http://en.wikibooks.org/wiki/Algebra ...
2.5 Web material 31 MatureLinearAlgebra.pdf http://www.bookrags.com/sciences/mathematics/vector-spaces-wom.html http://www.cap-l ...
32 2 Preliminary Linear Algebra http://www.math.poly.edu/courses/ma2012/Notes/GeneralLinearT.pdf http://www.math.psu.edu/xu/45l/ ...
3 Orthogonality In this chapter, we will analyze distance functions, inner products, projection and orthogonality, the process o ...
34 3 Orthogonality Furthermore, we know the following relations: y/n < x < loo — 12' l 2 < li<IMI^2 - y/n < X I & ...
3.1 Inner Products 35 Remark 3.1.5 \\x\\l geometrically amounts to the Pythagoras formula ap- plied (n-1) times. Definition 3.1. ...
36 3 Orthogonality Remark 3.1.13 The following statements are equivalent, i. W^V^1. a v = w ± Hi. W ± V and dimV + dimW — n. Pro ...
3.1 Inner Products 37 Proposition 3.1.16 The cosine of the angle between any two vectors u and v is COSC : U V T iu \\v\\ Remark ...
38 3 Orthogonality Proof. (ATA)T = AT{AT)T = ATA. Claim: N{A) = H{AT A). i. M{A) C M{ATA) : x e M(A) =• Ax = 6 =>• 4rAr -iT9 ...
3.2 Projections and Least Squares Approximations 39 Conversely, any matrix with the above two properties represents a projection ...
40 3 Orthogonality Pa=(a,b,0) Fig. 3.4. Orthogonal projection Remark 3.2.9 When we find an orthogonal basis that spans the groun ...
3.2 Projections and Least Squares Approximations 41 Proposition 3.2.11 Any set of independent vectors ai,a,2,- • • ,an can be co ...
42 3 Orthogonality Definition 3.2.14 A = QR is known as Q~R decomposition. Remark 3.2.15 If A = QR, then it is easy to solve Ax ...
3.2 Projections and Least Squares Approximations 43 => x = 0" L a 0 = A*b = "0 0 0 OiO ooi 0 0 0 Thus, A* = "0 0 0 0 .0 0 0" ...
44 3 Orthogonality 3.3 Summary for Ax = b Let us start with the simplest case which is illustrated in Figure 3.5. A G R"x" is sq ...
3.3 Summary for Ax = b 45 Fig. 3.6. Parametric solution: b 6 TZ(A), A : m x n, and r — rank(A) What if b $. 11(A)? We cannot fin ...
46 3 Orthogonality w Fig. 3.8. Least norm squared solution: (ATA) is not invertible and A^ = QiE'Qj Table 3.1. How to solve Ax = ...
3.4 Web material 47 the smallest magnitude, in some engineering applications. We may use the singular value decomposition in thi ...
48 3 Orthogonality Norm_Induced_Inner_Product.html http://ccrma-www.Stanford.edu/~jos/r320/Inner_Product.html http://ccrma-www.S ...
3.4 Web material 49 http://psblade.ucdavis.edu/papers/ginv.pdf http://public.lanl.gov/mewall/kluwer2002.html http://rkb.home.cer ...
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