Chap. 5 Inner Product Spaces 5.1 Length and Dot Product in R n 5.2 Inner Product Spaces 5.3 Orthonormal Bases: Gram-Schmidt Process 5.4 Mathematical Models

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Chap. 5 Inner Product Spaces 5.1 Length and Dot Product in R n 5.2 Inner Product Spaces 5.3 Orthonormal Bases: Gram-Schmidt Process 5.4 Mathematical Models & Least Squares Analysis 5.5 Applications of Inner Product Spaces Slide 2 Ming-Feng YehChapter 55-2 Vectors in the plane can be characterized as directed line segments having a certain length and direction. If v = (v 1, v 2 ), then the length, or magnitude, of v, denoted by, is defined to be The length or norm of a vector in R n is given by Unit vector: Zero vector: 5.1 Length and Dot Product Slide 3 Ming-Feng YehChapter 55-3 Standard Unit Vector Each vector in the standard basis for R n has length 1 and is called a standard unit vector. R 2 : {i, j} = {(1, 0), (0, 1)} R 3 : {i, j, k} = {(1, 0, 0), (0, 1, 0), (0, 0, 1)} (1, 1) = (1, 0) + (0, 1) = i + j (2, 2) = 2(1, 1) = 2i + 2j Section 5-1 (1, 1) (2, 2) = 2(1, 1) i j 0, then u and v have the same direction, and if c < 0, then u and v have the opposite direction. Theorem 5.1: Length of a Scalar Multiple Let v be a vector in R n and c be a scalar. Then where is the absolute value of c. Section 5-1"> Ming-Feng YehChapter 55-4 Length of a Scalar Multiple Two nonzero vectors u and v in R n are parallel if one is a scalar multiple of the other, i.e., u = cv. If c > 0, then u and v have the same direction, and if c < 0, then u and v have the opposite direction. Theorem 5.1: Length of a Scalar Multiple Let v be a vector in R n and c be a scalar. Then where is the absolute value of c. Section 5-1 Slide 5 Ming-Feng YehChapter 55-5 Theorem 5.2 Unit Vector in the Direction of v If v is a nonzero vector in R n, then the vector has length 1 and has the same direction as v. This vector u is called the unit vector in the direction of v. pf: 1. v is nonzero is positive and Therefore u has the same direction as v. 2. The above process is called normalizing the vector v. Section 5-1 Slide 6 Ming-Feng YehChapter 55-6 Example 2 Find the unit vector in the direction of v = (3, 1, 2), and verify that this vector has length 1. Sol: The unit vector in the direction of v is which is a unit vector because Section 5-1 Slide 7 Ming-Feng YehChapter 55-7 Distance Between 2 Vectors: R 2 The distance between two points in the plane, (u 1, u 2 ) and (v 1, v 2 ), is given by In vector terminology, this distance can be viewed as the length of u v, i.e., Section 5-1 Slide 8 Ming-Feng YehChapter 55-8 Distance Between 2 Vectors: R n The distance between two vectors u and v in R n is Properties of distance: 1. 2. if and only if u = v. 3. Let u = (0, 2, 2) and v = (2, 0, 1). Then Section 5-1 Slide 9 Ming-Feng YehChapter 55-9 Angle Between Two Vectors: R 2 The angle between two nonzero vector u = (u 1, u 2 ) and v = (v 1, v 2 ) is The Law of Cosines Section 5-1 Slide 10 Ming-Feng YehChapter 55-10 Dot Product The dot product of u = (u 1, u 2, , u n ) and v = (v 1, v 2, , v n ) is the scalar quantity Theorem 5.3: Properties of Dot Product 1. 2. 3. 4. 5. if and only if v = 0. Section 5-1 Slide 11 Ming-Feng YehChapter 55-11 Example 5 Given u = (2, 2), v = (5, 8) and w = ( 4, 3). 1. 2. 3. 4. 5. Section 5-1 Slide 12 Ming-Feng YehChapter 55-12 Example 6 Given two vectors u and v in R n such that, and evaluate Sol: Section 5-1 Slide 13 Ming-Feng YehChapter 55-13 Angle Between Two Vectors: R n The angle between two nonzero vectors u and v in R n is Two vectors u and v in R n is orthogonal if The zero vector 0 is orthogonal to every vector. Section 5-1 Slide 14 Ming-Feng YehChapter 55-14 Theorem 5.4 Cauchy-Schwarz Inequality If u and v are vectors in R n, then where denotes the absolute value of pf: If u = 0, then Hence the theorem is true if u = 0. If from and we have Section 5-1 Slide 15 Ming-Feng YehChapter 55-15 Example 7 Verify the Cauchy-Schwarz Inequality for u = (1, 1, 3) and v = (2, 0, 1). Sol: Because and, we have Therefore Section 5-1 Slide 16 Ming-Feng YehChapter 55-16 Examples 8 & 9 Ex 8: The angle between u = (4, 0, 2, 2) and v = (2, 0, 1, 1) is given by u and v should have opposite directions, because u = 2v. Ex 9: The vectors u = (3, 2, 1, 4) and v = (1, 1, 1, 0) are orthogonal because Section 5-1 Slide 17 Ming-Feng YehChapter 55-17 Example 10 Determine all vectors in R 2 that are orthogonal to u = (4, 2). Sol: Let v = (v 1, v 2 ) be orthogonal to u. Then This implies that Therefore every vector that is orthogonal to (4, 2) is of the form Section 5-1 Slide 18 Ming-Feng YehChapter 55-18 Theorem 5.5: Triangle Inequality If u and v are vectors in R n, then pf: Section 5-1 Equality occurs if and only if the vectors u and v have the same directions. Slide 19 Ming-Feng YehChapter 55-19 Theorem 5.6: Pythagorean Thm If u and v are vectors in R n, then u and v are orthogonal iff pf: If u and v are orthogonal, then Section 5-1 Slide 20 Ming-Feng YehChapter 55-20 Dot Product and Matrix Multiplication Represent a vector in R n as an n 1 column matrix. Let and. Then Section 5-1 Slide 21 Ming-Feng YehChapter 55-21 5.2 Inner Product Spaces = dot product (Euclidean inner product for R n ) = general inner product for vector space V. Definition: Let u, v, and w be vectors in a vector space V, and let c be any scalar. An inner product on V is a function that associates a real number with pair of vectors u and v and satisfies the following axioms. 1. 2. 3. 4. and iff v = 0. Slide 22 Ming-Feng YehChapter 55-22 Inner Product Space A vector space V with an inner product is called an inner product space. Show that the following function defines an inner product on R 2 : where and pf: 1. 2. Let. Then Section 5-2 Slide 23 Ming-Feng YehChapter 55-23 Proof of Theorem 2 3. If c is any scalar, then 4. Because the square of a real number is nonnegative, Moreover, this expression is equal to zero iff v = 0. Section 5-2 Slide 24 Ming-Feng YehChapter 55-24 Example 3 Show that the following function is not an inner product on R 3 : where and pf: Let v = (1, 2, 1). Then Axiom 4 is not satisfied. Section 5-2 Slide 25 Ming-Feng YehChapter 55-25 Inner Product on M 2,2 & P n Let and be matrices in the vector space M 2,2. The function given by is an inner product on M 2,2. Let and be polynomials in the vector space P n. The function given by is an inner product on P n. The verification of the four inner product axioms is left to you. Section 5-2 Slide 26 Ming-Feng YehChapter 55-26 Theorem 5.7 Properties of Inner Products Let u, v, and w be vectors in an inner product space V, and let c be any real number. 1. 2. 3. Section 5-2 Slide 27 Ming-Feng YehChapter 55-27 Norm, Distance & Angle Let u and v be vectors in an inner product space V. 1. The norm (or length) of u is 2. The distance between u and v is 3. The angle between two nonzero vectors u and v is given by 4. u and v are orthogonal if If, then v is called a unit vector. If v is any nonzero vector, then the vector is called the unit vector in the direction of v. Section 5-2 Slide 28 Ming-Feng YehChapter 55-28 Example 6 Let and be polynomial in P 2, and determine the following. 1. 2. 3. 4. Section 5-2 Slide 29 Ming-Feng YehChapter 55-29 Theorem 5.8 Let u and v be vectors in an inner product space V. 1. Cauchy-Schwarz Inequality: 2. Triangle Inequality: 3. Pythagorean Theorem: u and v are orthogonal iff Section 5-2 Slide 30 Ming-Feng YehChapter 55-30 Orthogonal Projections: R 2 Let u and v be vectors in the plane. If v is nonzero, then u can be orthogonally projected onto v. This projection is denoted by proj v u. proj v u is a scalar multiple of v, i.e., proj v u = av. If a > 0, then and Therefore u v proj v u Section 5-2 Slide 31 Ming-Feng YehChapter 55-31 Proj v u in R 2 & Example 9 If a < 0, then The orthogonal projection of u onto v is given by the same formula. Example 9: The orthogonal projection of u = (4, 2) onto v = (3, 4) is given by u v proj v u v u Section 5-2 Slide 32 Ming-Feng YehChapter 55-32 Orthogonal Projection & Ex 10 Let u and v be vectors in an inner product space V, such that Then the orthogonal projection of u onto v is given by Example 10: The orthogonal projection of u = (6, 2, 4) onto v = (1, 2, 0) is given by Section 5-2 Slide 33 Ming-Feng YehChapter 55-33 Remark of Example 10 In Example 10, u = (6, 2, 4), v = (1, 2, 0), and proj v u = (2, 4, 0). u proj v u = (6, 2, 4) (2, 4, 0) = (4, 2, 4) is orthogonal to v = (1, 2, 0). If u and v are nonzero vectors in an inner product space, then u proj v u is orthogonal to v. u v proj v u d(u, proj v u) Section 5-2 Slide 34 Ming-Feng YehChapter 55-34 Theorem 5.9 Orthogonal Projection and Distance Let u and v be vectors in an inner product space V, such that Then u v proj v u d(u, cv) d(u, proj v u) Section 5-2 Slide 35 Ming-Feng YehChapter 55-35 5.3 Orthogonal Bases The standard basis for R 3 : B = {(1, 0, 0), (0, 1, 0), (0, 0, 1)} 1. The three vectors are mutually orthogonal. 2. Each vector in the basis is a unit vector. Definition: Orthogonal & Orthonormal Sets A set S of vectors in an inner product space V is called orthogonal if every pair of vectors in S is orthogonal. If, in addition, each vector in the set is a unit vector, then S is called orthonomal. Slide 36 Ming-Feng YehChapter 55-36 Orthonormal Basis For S = {v 1, v 2, , v n }, Orthogonal Orthonormal 1. = 0, i j 1. = 0, i j 2. If S is a basis, then it is called an orthogonal basis or an orthonormal basis. The standard basis for R n is orthomormal, but it is not the only orthonormal basis for R n. For example, is a nonstandard orthonormal basis for R 3. Section 5-3 Slide 37 Ming-Feng YehChapter 55-37 Examples 1 & 2 Example 1: Show that the following set is an orthonormal basis for R 3. Sol: 1. 2. Therefore S is an orthonormal set. Example 2: In P 3, with inner product = a 0 b 0 + a 1 b 1 + a 2 b 2 + a 3 b 3, the standard basis B = {1, x, x 2, x 3 } is orthonormal Section 5-3 Slide 38 Ming-Feng YehChapter 55-38 Theorem 5.10 Orthogonal Sets Are Linearly Independent If S = {v 1,