LA07_5 -...

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Unformatted text preview: ___________________________________________________________ Linear Algebra: Spring 2007 __________________________________________________________________________________ Proprietary of SAS Lab., Hanyang University - Ansan 5-1 Chapter 5. Orthogonality ♣ Geometric interpretation of matrices and vectors 1. Orthogonality in n-D vector space Consider a set of standard vectors in a coordinate system, { e 1 , e 2 , …, e n }. They possess a couple of good properties; orthogonality and unit length. Definition Two vectors v and w in n ℜ are said to be orthogonal each other, if = • w v . A set of vectors { v 1 , v 2 , …, v n } in n ℜ is said to be an orthogonal set if all pairs of distinct vectors are orthogonal (pairwise orthogonal). ╬ A set of standard vectors { e 1 , e 2 , …, e n } in n ℜ is an orthogonal set. ╬ Two vectors are orthononal , if they are orthogonal and of unit length. ╬ Two orthogonal vectors are perpendicular in geometrical sense. Theorem 5.1 If { v 1 , v 2 , …, v n } is an orthogonal set of non-zero vectors in n ℜ , then they are linearly independent. ← To show the independence, consider v v v = + + + n n c c c L 2 2 1 1 Definition An orthogonal basis for a subspace W of n ℜ is a basis of W that is an orthogonal set. ← A set of standard vectors { e 1 , e 2 , …, e n } in n ℜ is an orthogonal set. For example, (1) Vectors ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − = 1 1 2 1 v , ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − = 1 1 2 v , ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − = 1 1 1 3 v (2) An orthogonal basis for the subspace W of 3 ℜ given by ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ = + − ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = 2 ; z y x z y x W (a plane through the origin in 3 ℜ ) Note the vectors ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = 1 1 u and ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡− = 1 2 v are a basis for W (not orthogonal, though). To find a orthogonal basis, we need to find an extra vector W ∈ w , s.t. = • u w . ___________________________________________________________ Linear Algebra: Spring 2007 __________________________________________________________________________________ Proprietary of SAS Lab., Hanyang University - Ansan 5-2 Theorem 5.2 If { v 1 , v 2 , …, v k } is an orthogonal basis for a subspace W of n ℜ and let w be any vector in W . Then there exists a unique set of scalars { c 1 , c 2 , …, c k }, s.t. k k c c c v v v w + + + = L 2 2 1 1 , where i i i i c v v v w • • = , for i=1, …, k. ← It suffices to show that c i ’s are unique. Definition An ) ( n n × matrix Q is called an orthogonal matrix , if its columns form an orthonormal set. ╬ If Q is an orthogonal matrix, then Q T Q = I n ....
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LA07_5 -...

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