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orthogonality

# orthogonality - Orthogonality Tutorial Page 1 of 7...

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TUTORIALS TUTORIALS HOME GENERAL MATH NOTATION & METHODS OF PROOF INDUCTION COMPLEX NUMBERS POLYNOMIALS LINEAR ALGEBRA VECTORS SYSTEM OF LINEAR EQUATIONS MATRICES EIGENVALUES & EIGENVECTORS ORTHOGONALITY VECTOR SPACE DISTANCE & APPROXIMATION HOME TESTS TUTORIALS SAMPLE PROBLEMS COMMON MISTAKES STUDY TIPS GLOSSARY APPLICATIONS MATH HUMOUR ORTHOGONALITY TUTORIAL Orthogonal Sets Vectors v , u are orthogonal or perpendicular to each other if v u = 0 whenever v u . We say a set of vectors { v 1 , v 2 , ... , v k } is an orthogonal set if for all v j and v i , v j v i = 0 where i j and i, j = 1, 2, ... , k We can show easily that the standard basis in is an orthogonal set This is also true for any subset of the standard basis. Example one checks some other vectors for orthogonality. Next we will look at some theorems that apply to orthogonal sets. Theorem 1: If we have an orthogonal set { v 1 , v 2 , ... , v k } in then vectors v 1 , v 2 , ... , v k are all linearly independent. 1 | Proof Theorem 2: An orthogonal set of vectors v 1 , v 2 , ... , v n in form an orthogonal basis of Theorem 3: Letting { v 1 , v 2 , ... , v k } be an orthogonal basis for a subspace of and let w be any vector in the subspace . Then there exists scalars c 1 , c 2 , ... , c i , ... , c k such that: w = c 1 v 1 +c 2 v 2 + ... + c i v i + ... +c k v k Examples: 1 | Check vectors for orthogonality 2 | Find coordinates with respect to a basis Page 1 of 7 Orthogonality Tutorial 10/10/2011 http://www.nipissingu.ca/algebra/tutorials/orthogonality.html

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Orthonormal Sets Next we will begin to explain the difference between an orthogonal and an orthonormal set. The length or magnitude of v in is defined as || v || = A unit vector u in is a vector that has a length or magnitude of one. In other words: || u || = 1 or u u = 1 An orthonormal set is a set of vectors { v 1 , v 2 , ... , v k } which is orthogonal and every vector in the set is a unit vector. In other words for every vector v i in the set: || v i || = 1 and where i = 1, 2, ... , k It is quick and easy to obtain an orthonormal set for vectors from an orthogonal set of vectors, simply divide each vector by its length.
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orthogonality - Orthogonality Tutorial Page 1 of 7...

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