Page 1 Inner product spaces, Orthogonal and Orthonormal Vectors Institute of Lifelong Learning, University of Delhi Lesson: Inner product spaces, Orthogonal and Orthonormal Vectors Lesson Developer : Umesh Chand Department / College: Department of Mathematics, Kirorimal College, University of Delhi Page 2 Inner product spaces, Orthogonal and Orthonormal Vectors Institute of Lifelong Learning, University of Delhi Lesson: Inner product spaces, Orthogonal and Orthonormal Vectors Lesson Developer : Umesh Chand Department / College: Department of Mathematics, Kirorimal College, University of Delhi Inner product spaces, Orthogonal and Orthonormal Vectors Institute of Lifelong Learning, University of Delhi pg. 2 Table of Contents: Chapter: Inner product spaces, Orthogonal and Orthonormal Vectors 1. Learning outcomes 2. Introduction 3. Inner product space 4. Norm 5. Orthogonaility 6. Orthogonal complement ? Summary ? Exercise ? References/Bibliography/Further reading Page 3 Inner product spaces, Orthogonal and Orthonormal Vectors Institute of Lifelong Learning, University of Delhi Lesson: Inner product spaces, Orthogonal and Orthonormal Vectors Lesson Developer : Umesh Chand Department / College: Department of Mathematics, Kirorimal College, University of Delhi Inner product spaces, Orthogonal and Orthonormal Vectors Institute of Lifelong Learning, University of Delhi pg. 2 Table of Contents: Chapter: Inner product spaces, Orthogonal and Orthonormal Vectors 1. Learning outcomes 2. Introduction 3. Inner product space 4. Norm 5. Orthogonaility 6. Orthogonal complement ? Summary ? Exercise ? References/Bibliography/Further reading Inner product spaces, Orthogonal and Orthonormal Vectors Institute of Lifelong Learning, University of Delhi pg. 3 1. Learning outcomes After studying the whole content student will be able to understand ? Inner product space ? Norm ? Orthogonaility ? Orthogonal complement ? Schwarz inequality ? Bessel inequality 2. Introduction In linear algebra, an inner product space is vector space with additional structure called inner product. This additional structure associates each pair of vectors in the space with a scalar quantity know as the inner product of vectors. Inner products allow the rigorous introduction of intuitive geometrical notions such as the length of a vector or angle between two vectors. Inner product space generalize Euclidean space (in which the inner product is the det product, also known as scalar product) to vector spaces of any (possibility infinite) functional analysis. Page 4 Inner product spaces, Orthogonal and Orthonormal Vectors Institute of Lifelong Learning, University of Delhi Lesson: Inner product spaces, Orthogonal and Orthonormal Vectors Lesson Developer : Umesh Chand Department / College: Department of Mathematics, Kirorimal College, University of Delhi Inner product spaces, Orthogonal and Orthonormal Vectors Institute of Lifelong Learning, University of Delhi pg. 2 Table of Contents: Chapter: Inner product spaces, Orthogonal and Orthonormal Vectors 1. Learning outcomes 2. Introduction 3. Inner product space 4. Norm 5. Orthogonaility 6. Orthogonal complement ? Summary ? Exercise ? References/Bibliography/Further reading Inner product spaces, Orthogonal and Orthonormal Vectors Institute of Lifelong Learning, University of Delhi pg. 3 1. Learning outcomes After studying the whole content student will be able to understand ? Inner product space ? Norm ? Orthogonaility ? Orthogonal complement ? Schwarz inequality ? Bessel inequality 2. Introduction In linear algebra, an inner product space is vector space with additional structure called inner product. This additional structure associates each pair of vectors in the space with a scalar quantity know as the inner product of vectors. Inner products allow the rigorous introduction of intuitive geometrical notions such as the length of a vector or angle between two vectors. Inner product space generalize Euclidean space (in which the inner product is the det product, also known as scalar product) to vector spaces of any (possibility infinite) functional analysis. Inner product spaces, Orthogonal and Orthonormal Vectors Institute of Lifelong Learning, University of Delhi pg. 4 3. Inner Product Space Let V be a vector space over F. An inner product on V is a function V ?V to f that assign every ordered pair of vectors x, y in V, to a scalar in F, denoted by x, y , in such a way that (i) x y,z x,z y,z x,y,z V ? ? ? ? ? (ii) cx,y c x,y c F ? ? ? and x, y ? V (iii) x,y y,x x,y V ? ? ? (iv) x,x 0 if x 0 ?? Then the vector space V is said to be inner product space with respect to the specified inner product defined on it. Value Addition : Note 1. (i) and (ii) are equivalent to x y, z x,z y,z , F, x,y,z V ? ? ? ? ? ? ? ? ? ? ? ? 2. (i) and (ii) imply inner product is linear in Ist component, property (1) and (ii) Jointly is called linearly property, (iii) is called conjugate symmetry, (iv) is called non-negative. Example 1: Let V = F n , F = R (or C) Let x = (a 1 , a 2 , …, a n ), ? V y ? (b 1 , b 2 , …,b n ). Page 5 Inner product spaces, Orthogonal and Orthonormal Vectors Institute of Lifelong Learning, University of Delhi Lesson: Inner product spaces, Orthogonal and Orthonormal Vectors Lesson Developer : Umesh Chand Department / College: Department of Mathematics, Kirorimal College, University of Delhi Inner product spaces, Orthogonal and Orthonormal Vectors Institute of Lifelong Learning, University of Delhi pg. 2 Table of Contents: Chapter: Inner product spaces, Orthogonal and Orthonormal Vectors 1. Learning outcomes 2. Introduction 3. Inner product space 4. Norm 5. Orthogonaility 6. Orthogonal complement ? Summary ? Exercise ? References/Bibliography/Further reading Inner product spaces, Orthogonal and Orthonormal Vectors Institute of Lifelong Learning, University of Delhi pg. 3 1. Learning outcomes After studying the whole content student will be able to understand ? Inner product space ? Norm ? Orthogonaility ? Orthogonal complement ? Schwarz inequality ? Bessel inequality 2. Introduction In linear algebra, an inner product space is vector space with additional structure called inner product. This additional structure associates each pair of vectors in the space with a scalar quantity know as the inner product of vectors. Inner products allow the rigorous introduction of intuitive geometrical notions such as the length of a vector or angle between two vectors. Inner product space generalize Euclidean space (in which the inner product is the det product, also known as scalar product) to vector spaces of any (possibility infinite) functional analysis. Inner product spaces, Orthogonal and Orthonormal Vectors Institute of Lifelong Learning, University of Delhi pg. 4 3. Inner Product Space Let V be a vector space over F. An inner product on V is a function V ?V to f that assign every ordered pair of vectors x, y in V, to a scalar in F, denoted by x, y , in such a way that (i) x y,z x,z y,z x,y,z V ? ? ? ? ? (ii) cx,y c x,y c F ? ? ? and x, y ? V (iii) x,y y,x x,y V ? ? ? (iv) x,x 0 if x 0 ?? Then the vector space V is said to be inner product space with respect to the specified inner product defined on it. Value Addition : Note 1. (i) and (ii) are equivalent to x y, z x,z y,z , F, x,y,z V ? ? ? ? ? ? ? ? ? ? ? ? 2. (i) and (ii) imply inner product is linear in Ist component, property (1) and (ii) Jointly is called linearly property, (iii) is called conjugate symmetry, (iv) is called non-negative. Example 1: Let V = F n , F = R (or C) Let x = (a 1 , a 2 , …, a n ), ? V y ? (b 1 , b 2 , …,b n ). Inner product spaces, Orthogonal and Orthonormal Vectors Institute of Lifelong Learning, University of Delhi pg. 5 Define n 1 1 2 2 n n i i i1 x, y a b a b .... a b a b ? ? ? ? ? ? ? Then it is an inner product on V, called standard inner product. Verification: Let x, y, z ? V, where x = (a 1 , a 2 , …, a n ), y = (b 1 , b 2 , ….,.b n ), z = (c, c 2 , …, c n ) (i) n i i i i1 x y,z (a b )c ? ? ? ? ? nn i i i i i 1 i 1 a c b c ?? ?? ?? x,z y,z ?? (ii) n ii i1 cx, y ca b ? ? ? n ii i1 c a b ? ? ? c x, y ? (iii) n ii i1 x, y a b ? ? ? n ii i1 ab ? ? ? n ii i1 ba ? ? ? y,x ? (iv) n ii i1 x,x a a 0 ? ?? ? if x ? 0 Thus x, y is an inner product on V.Read More

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