14-orthogonal

14-orthogonal - ORTHOGONALITY Math 21b, O. Knill HOMEWORK:...

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Unformatted text preview: ORTHOGONALITY Math 21b, O. Knill HOMEWORK: Section 5.1: 6,10,16,20,28,38*,14* ORTHOGONALITY. ~v and ~w are called orthogonal if ~v ~w = 0. Examples. 1) 1 2 and 6- 3 are orthogonal in R 2 . 2) ~v and w are both orthogonal to ~v ~w in R 3 . ~v is called a unit vector if || ~v || = ~v ~v = 1. B = { ~v 1 , . . . , ~v n } are called orthogonal if they are pairwise orthogonal. They are called orthonormal if they are also unit vectors. A basis is called an orthonormal basis if it is orthonormal. For an orthonormal basis, the matrix A ij = ~v i ~v j is the unit matrix. FACT. Orthogonal vectors are linearly independent and n orthogonal vectors in R n form a basis. Proof. The dot product of a linear relation a 1 ~v 1 + . . . + a n ~v n = 0 with ~v k gives a k ~v k ~v k = a k || ~v k || 2 = 0 so that a k = 0. If we have n linear independent vectors in R n then they automatically span the space. ORTHOGONAL COMPLEMENT. A vector ~w R n is called orthogonal to a linear space V if ~w is orthogonal to every vector in ~v V . The orthogonal complement of a linear space V is the set W of all vectors which are orthogonal to V . It forms a linear space because ~v ~w 1 = 0 , ~v ~w 2 = 0 implies ~v ( ~w 1 + ~w 2 ) = 0. ORTHOGONAL PROJECTION. The orthogonal projection onto a linear space V with orthnormal basis ~v 1 , . . . , ~v n is the linear map T ( ~x ) = proj V ( x ) = ( ~v 1 ~x ) ~v 1 + . . . + ( ~v n ~x ) ~v n The vector ~x- proj V ( ~x ) is in the orthogonal complement of V . (Note that ~v i in the projection formula are unit vectors, they have also to be orthogonal.) SPECIAL CASE. For an orthonormal basis ~v i , one can write ~x = ( ~v 1 ~x ) ~v 1 + ... + ( ~v n ~x ) ~v n ....
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This note was uploaded on 04/06/2008 for the course MATH 21B taught by Professor Judson during the Spring '03 term at Harvard.

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