DOCSLIB.ORG

Lecture 23: 6.1 Inner Products

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lecture 23: 6.1 Inner Products Lecture 23: 6.1 Inner Products Wei-Ta Chu 2008/12/17 Definition An inner product on a real vector space V is a function that associates a real number u, vwith each pair of vectors u and v in V in such a way that the following axioms are satisfied for all vectors u, v, and w in V and all scalars k. u, v= v, u u + v, w= u, w+ v, w ku, v= k u, v u, u0 and u, u=0if andonlyif u = 0 A real vector space with an inner product is called a real inner product space. 2008/12/17 Elementary Linear Algebra 2 Euclidean Inner Product on Rn If u = (u1, u2, …, un) and v = (v1, v2, …, vn) are vectors in Rn, then the formula v, u= u · v = u1v1 + u2v2 + … + unvn defines v, uto be the Euclidean product on Rn. The four inner product axioms hold by Theorem 4.1.2. 2008/12/17 Elementary Linear Algebra 3 Properties of Euclidean Inner Product Theorem 4.1.2 n If u, v and w are vectors in R and k is any scalar, then u · v = v · u (u + v) · w = u · w + v · w (k u) · v = k (u · v) v · v ≥0; Further, v · v = 0 if and only if v = 0 Example (3u + 2v) · (4u + v) = (3u) · (4u + v) + (2v) · (4u + v ) = (3u) · (4u) + (3u) · v + (2v) · (4u) + (2v) · v =12(u · u) + 11(u · v) + 2(v · v) 2008/12/17 Elementary Linear Algebra 4 Euclidean Inner Product vs. Inner Product The Euclidean inner product is the most important inner product on Rn. However, there are various applications in which it is desirable to modify the Euclidean inner product by weighting its terms differently. More precisely, if w1,w2,…,wn are positive real numbers, and if u=(u1,u2,…,un) and v=(v1,v2,…,vn) are vectors in Rn, then it can be shown which is called weighted Euclidean inner product with weights w1,w2,…,wn … 2008/12/17 Elementary Linear Algebra 5 Example Suppose that some physical experiment can produce any of n possible numerical values x1,x2,…,xn. We perform m repetitions of the experiment and yield these values with various frequencies; that is, x1 occurs f1 times, x2 occurs f2 times, and so forth. f1+f2+…+fn=m. Thus, the arithmetic average, or mean, is 2008/12/17 Elementary Linear Algebra 6 Example If we let f=(f1,f2,…,fn), x=(x1,x2,…,xn), w1=w2=…=wn=1/m Then this equation can be expressed as the weighted inner product 2008/12/17 Elementary Linear Algebra 7 Weighted Euclidean Product 2 Let u = (u1, u2) and v = (v1, v2) be vectors in R . Verify that the weighted Euclidean inner product u, v= 3u1v1 + 2u2v2 satisfies the four product axioms. Solution: Note first that if u and v are interchanged in this equation, the right side remains the same. Therefore, u, v= v, u. If w = (w1, w2), then u + v, w= 3(u1+v1)w1 + 2(u2+v2)w2 = (3u1w1 + 2u2w2) + (3v1w1 + 2v2w2)= u, w+ v, w which establishes the second axiom. 2008/12/17 Elementary Linear Algebra 8 Weighted Euclidean Product ku, v=3(ku1)v1 + 2(ku2)v2 = k(3u1v1 + 2u2v2) = k u, v which establishes the third axiom. 2 2 2 v, v= 3v1v1+2v2v2 = 3v1 + 2v2 .Obviously , v, v= 3v1 + 2 2 2 2v2 ≥0 . Furthermore, v, v= 3v1 + 2v2 = 0 if and only if v1 = v2 = 0, That is , if and only if v = (v1,v2)=0. Thus, the fourth axiom is satisfied. 2008/12/17 Elementary Linear Algebra 9 Definition If V is an inner product space, then the norm (or length) of a vector u in V is denoted by ||u|| and is defined by ||u|| = u, u½ The distance between two points (vectors) u and v is denoted by d(u,v) and is defined by d(u, v) = ||u –v|| If a vector has norm 1, then we say that it is a unit vector. 2008/12/17 Elementary Linear Algebra 10 Norm and Distance in Rn If u = (u1, u2, …, un) and v = (v1, v2, …, vn) are vectors in Rn with the Euclidean inner product, then d(,),()()uv uv uvuv 1/ 2 uvuv 1/ 2 2 2 2 (u1 v 1 ) ( u 2 v 2 ) ... ( un v n ) 2008/12/17 Elementary Linear Algebra 11 Weighted Euclidean Inner Product The norm and distance depend on the inner product used. If the inner product is changed, then the norms and distances between vectors also change. 2 For example, for the vectors u = (1,0) and v = (0,1) in R with the Euclidean inner product, we have u 1 and d(u, v) u v (1,1) 12 (1)2 2 However, if we change to the weighted Euclidean inner product u, v= 3u1v1 + 2u2v2 , then we obtain u u,u 1/ 2 (311200)1/ 2 3 d(u, v) u v (1,1),(1,1) 1/ 2 [3112(1)(1)]1/ 2 5 2008/12/17 Elementary Linear Algebra 12 Unit Circles and Spheres in Inner Product Space If V is an inner product space, then the set of points in V that satisfy ||u|| = 1 is called the unite sphere or sometimes the unit circle in V. In R2 and R3 these are the points that lie 1 unit away form the origin. 2008/12/17 Elementary Linear Algebra 13 Unit Circles in R2 Sketch the unit circle in an xy-coordinate system in R2 using the Euclidean inner product u, v= u1v1 + u2v2 Sketch the unit circle in an xy-coordinate system in R2 using the Euclidean inner product u, v= 1/9u1v1 + 1/4u2v2 Solution ½ 2 2 ½ If u = (x,y), then ||u|| = u, u = (x + y ) , so the equation of the unit circle is x2 + y2 = 1. ½ 2 2 ½ If u = (x,y), then ||u|| = u, u = (1/9x + 1/4y ) , so the equation of the unit circle is x2 /9 + y2/4 = 1. 2008/12/17 Elementary Linear Algebra 14 Inner Products Generated by Matrices u1 v 1 u2 v 2 Letu and v = be vectors in Rn (expressed as n1 :: un v n matrices), and let A be an invertible nn matrix. If u · v is the Euclidean inner product on Rn, then the formula u, v= Au · Av defines an inner product; it is called the inner product on Rn generated by A. Recalling that the Euclidean inner product u · v can be written as the matrix product vTu, the above formula can be written in the alternative form u, v= (Av) TAu, or equivalently, u, v= vTATAu 2008/12/17 Elementary Linear Algebra 15 Inner Product Generated by the Identity Matrix The inner product on Rn generated by the nn identity matrix is the Euclidean inner product: Let A = I, we have u, v= Iu · Iv = u · v The weighted Euclidean inner product u, v= 3u1v1 + 2u2v2 is the inner product on R2 generated by 3 0 A 0 2 since 3 0 3 0 u1 3 0u1 u,v v1 v2 v1 v2 3u1v1 2u2v2 0 20 2u2 0 2u2 2008/12/17 Elementary Linear Algebra 16 Inner Product Generated by the Identity Matrix In general, the weighted Euclidean inner product u, v= w1u1v1 + w2u2v2 + … + wnunvn is the inner product on Rn generated by 2008/12/17 Elementary Linear Algebra 17 An Inner Product on M22 u1 u 2 v 1 v 2 If UV and u3 u 4 v 3 v 4 are any two 22 matrices, then T T U, V= tr(U V) = tr(V U) = u1v1 + u2v2 + u3v3 + u4v4 defines an inner product on M22 For example, if 1 2 1 0 UV and 3 4 3 2 then U, V= 1(-1)+2(0) + 3(3) + 4(2) = 16 The norm of a matrix U relative to this inner product is 1/ 2 2 2 2 2 U U, U u1 u 2 u 3 u 4 and the unit sphere in this space consists of all 22 matrices U whose entries satisfy the equation ||U|| = 1, which on squaring yields 2 2 2 2 u1 + u2 + u3 + u4 = 1 2008/12/17 Elementary Linear Algebra 18 An Inner Product on P2 2 2 If p = a0 + a1x + a2x and q = b0 + b1x + b2x are any two vectors in P2, then the following formula defines an inner product on P2: p, q= a0b0 + a1b1 + a2b2 The norm of the polynomial p relative to this inner product is 1/ 2 2 2 2 p p, p a0 a 1 a 2 and the unit sphere in this space consists of all polynomials p in P2 whose coefficients satisfy the equation || p || = 1, which on squaring yields 2 2 2 a0 + a1 + a2 =1 2008/12/17 Elementary Linear Algebra 19 Theorem 6.1.1 (Properties of Inner Products) If u, v, and w are vectors in a real inner product space, and k is any scalar, then: 0, v= v, 0= 0 u, v + w= u, v+ u, w u, kv=k u, v u –v, w= u, w–v, w u, v –w= u, v–u, w 2008/12/17 Elementary Linear Algebra 20 Example u –2v, 3u + 4v = u, 3u + 4v–2v, 3u+4v = u, 3u+ u, 4v–2v, 3u–2v, 4v = 3 u, u+ 4 u, v–6 v, u–8 v, v = 3 || u ||2 + 4 u, v–6 u, v–8 || v ||2 = 3 || u ||2 –2 u, v–8 || v ||2 2008/12/17 Elementary Linear Algebra 21 Example We are guaranteed without any further proof that the five properties given in Theorem 6.1.1 are true for the inner product on Rn generated by any matrix A.
Load more

Recommended publications
  • Introduction to Linear Bialgebra
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by University of New Mexico University of New Mexico UNM Digital Repository Mathematics and Statistics Faculty and Staff Publications Academic Department Resources 2005 INTRODUCTION TO LINEAR BIALGEBRA Florentin Smarandache University of New Mexico, [email protected] W.B. Vasantha Kandasamy K. Ilanthenral Follow this and additional works at: https://digitalrepository.unm.edu/math_fsp Part of the Algebra Commons, Analysis Commons, Discrete Mathematics and Combinatorics Commons, and the Other Mathematics Commons Recommended Citation Smarandache, Florentin; W.B. Vasantha Kandasamy; and K. Ilanthenral. "INTRODUCTION TO LINEAR BIALGEBRA." (2005). https://digitalrepository.unm.edu/math_fsp/232 This Book is brought to you for free and open access by the Academic Department Resources at UNM Digital Repository. It has been accepted for inclusion in Mathematics and Statistics Faculty and Staff Publications by an authorized administrator of UNM Digital Repository. For more information, please contact [email protected], [email protected], [email protected]. INTRODUCTION TO LINEAR BIALGEBRA W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Madras Chennai – 600036, India e-mail: [email protected] web: http://mat.iitm.ac.in/~wbv Florentin Smarandache Department of Mathematics University of New Mexico Gallup, NM 87301, USA e-mail: [email protected] K. Ilanthenral Editor, Maths Tiger, Quarterly Journal Flat No.11, Mayura Park, 16, Kazhikundram Main Road, Tharamani, Chennai – 600 113, India e-mail: [email protected] HEXIS Phoenix, Arizona 2005 1 This book can be ordered in a paper bound reprint from: Books on Demand ProQuest Information & Learning (University of Microfilm International) 300 N.
    [Show full text]
  • 21. Orthonormal Bases
    21. Orthonormal Bases The canonical/standard basis 011 001 001 B C B C B C B0C B1C B0C e1 = B.C ; e2 = B.C ; : : : ; en = B.C B.C B.C B.C @.A @.A @.A 0 0 1 has many useful properties. • Each of the standard basis vectors has unit length: q p T jjeijj = ei ei = ei ei = 1: • The standard basis vectors are orthogonal (in other words, at right angles or perpendicular). T ei ej = ei ej = 0 when i 6= j This is summarized by ( 1 i = j eT e = δ = ; i j ij 0 i 6= j where δij is the Kronecker delta. Notice that the Kronecker delta gives the entries of the identity matrix. Given column vectors v and w, we have seen that the dot product v w is the same as the matrix multiplication vT w. This is the inner product on n T R . We can also form the outer product vw , which gives a square matrix. 1 The outer product on the standard basis vectors is interesting. Set T Π1 = e1e1 011 B C B0C = B.C 1 0 ::: 0 B.C @.A 0 01 0 ::: 01 B C B0 0 ::: 0C = B. .C B. .C @. .A 0 0 ::: 0 . T Πn = enen 001 B C B0C = B.C 0 0 ::: 1 B.C @.A 1 00 0 ::: 01 B C B0 0 ::: 0C = B. .C B. .C @. .A 0 0 ::: 1 In short, Πi is the diagonal square matrix with a 1 in the ith diagonal position and zeros everywhere else.
    [Show full text]
  • Determinant Notes
    68 UNIT FIVE DETERMINANTS 5.1 INTRODUCTION In unit one the determinant of a 2× 2 matrix was introduced and used in the evaluation of a cross product. In this chapter we extend the definition of a determinant to any size square matrix. The determinant has a variety of applications. The value of the determinant of a square matrix A can be used to determine whether A is invertible or noninvertible. An explicit formula for A–1 exists that involves the determinant of A. Some systems of linear equations have solutions that can be expressed in terms of determinants. 5.2 DEFINITION OF THE DETERMINANT a11 a12 Recall that in chapter one the determinant of the 2× 2 matrix A = was a21 a22 defined to be the number a11a22 − a12 a21 and that the notation det (A) or A was used to represent the determinant of A. For any given n × n matrix A = a , the notation A [ ij ]n×n ij will be used to denote the (n −1)× (n −1) submatrix obtained from A by deleting the ith row and the jth column of A. The determinant of any size square matrix A = a is [ ij ]n×n defined recursively as follows. Definition of the Determinant Let A = a be an n × n matrix. [ ij ]n×n (1) If n = 1, that is A = [a11], then we define det (A) = a11 . n 1k+ (2) If na>=1, we define det(A) ∑(-1)11kk det(A ) k=1 Example If A = []5 , then by part (1) of the definition of the determinant, det (A) = 5.
    [Show full text]
  • Linear Algebra for Dummies
    Linear Algebra for Dummies Jorge A. Menendez October 6, 2017 Contents 1 Matrices and Vectors1 2 Matrix Multiplication2 3 Matrix Inverse, Pseudo-inverse4 4 Outer products 5 5 Inner Products 5 6 Example: Linear Regression7 7 Eigenstuff 8 8 Example: Covariance Matrices 11 9 Example: PCA 12 10 Useful resources 12 1 Matrices and Vectors An m × n matrix is simply an array of numbers: 2 3 a11 a12 : : : a1n 6 a21 a22 : : : a2n 7 A = 6 7 6 . 7 4 . 5 am1 am2 : : : amn where we define the indexing Aij = aij to designate the component in the ith row and jth column of A. The transpose of a matrix is obtained by flipping the rows with the columns: 2 3 a11 a21 : : : an1 6 a12 a22 : : : an2 7 AT = 6 7 6 . 7 4 . 5 a1m a2m : : : anm T which evidently is now an n × m matrix, with components Aij = Aji = aji. In other words, the transpose is obtained by simply flipping the row and column indeces. One particularly important matrix is called the identity matrix, which is composed of 1’s on the diagonal and 0’s everywhere else: 21 0 ::: 03 60 1 ::: 07 6 7 6. .. .7 4. .5 0 0 ::: 1 1 It is called the identity matrix because the product of any matrix with the identity matrix is identical to itself: AI = A In other words, I is the equivalent of the number 1 for matrices. For our purposes, a vector can simply be thought of as a matrix with one column1: 2 3 a1 6a2 7 a = 6 7 6 .
    [Show full text]
  • Determinants Math 122 Calculus III D Joyce, Fall 2012
    Determinants Math 122 Calculus III D Joyce, Fall 2012 What they are. A determinant is a value associated to a square array of numbers, that square array being called a square matrix. For example, here are determinants of a general 2 × 2 matrix and a general 3 × 3 matrix. a b = ad − bc: c d a b c d e f = aei + bfg + cdh − ceg − afh − bdi: g h i The determinant of a matrix A is usually denoted jAj or det (A). You can think of the rows of the determinant as being vectors. For the 3×3 matrix above, the vectors are u = (a; b; c), v = (d; e; f), and w = (g; h; i). Then the determinant is a value associated to n vectors in Rn. There's a general definition for n×n determinants. It's a particular signed sum of products of n entries in the matrix where each product is of one entry in each row and column. The two ways you can choose one entry in each row and column of the 2 × 2 matrix give you the two products ad and bc. There are six ways of chosing one entry in each row and column in a 3 × 3 matrix, and generally, there are n! ways in an n × n matrix. Thus, the determinant of a 4 × 4 matrix is the signed sum of 24, which is 4!, terms. In this general definition, half the terms are taken positively and half negatively. In class, we briefly saw how the signs are determined by permutations.
    [Show full text]
  • Math 2331 – Linear Algebra 6.2 Orthogonal Sets
    6.2 Orthogonal Sets Math 2331 { Linear Algebra 6.2 Orthogonal Sets Jiwen He Department of Mathematics, University of Houston [email protected] math.uh.edu/∼jiwenhe/math2331 Jiwen He, University of Houston Math 2331, Linear Algebra 1 / 12 6.2 Orthogonal Sets Orthogonal Sets Basis Projection Orthonormal Matrix 6.2 Orthogonal Sets Orthogonal Sets: Examples Orthogonal Sets: Theorem Orthogonal Basis: Examples Orthogonal Basis: Theorem Orthogonal Projections Orthonormal Sets Orthonormal Matrix: Examples Orthonormal Matrix: Theorems Jiwen He, University of Houston Math 2331, Linear Algebra 2 / 12 6.2 Orthogonal Sets Orthogonal Sets Basis Projection Orthonormal Matrix Orthogonal Sets Orthogonal Sets n A set of vectors fu1; u2;:::; upg in R is called an orthogonal set if ui · uj = 0 whenever i 6= j. Example 82 3 2 3 2 39 < 1 1 0 = Is 4 −1 5 ; 4 1 5 ; 4 0 5 an orthogonal set? : 0 0 1 ; Solution: Label the vectors u1; u2; and u3 respectively. Then u1 · u2 = u1 · u3 = u2 · u3 = Therefore, fu1; u2; u3g is an orthogonal set. Jiwen He, University of Houston Math 2331, Linear Algebra 3 / 12 6.2 Orthogonal Sets Orthogonal Sets Basis Projection Orthonormal Matrix Orthogonal Sets: Theorem Theorem (4) Suppose S = fu1; u2;:::; upg is an orthogonal set of nonzero n vectors in R and W =spanfu1; u2;:::; upg. Then S is a linearly independent set and is therefore a basis for W . Partial Proof: Suppose c1u1 + c2u2 + ··· + cpup = 0 (c1u1 + c2u2 + ··· + cpup) · = 0· (c1u1) · u1 + (c2u2) · u1 + ··· + (cpup) · u1 = 0 c1 (u1 · u1) + c2 (u2 · u1) + ··· + cp (up · u1) = 0 c1 (u1 · u1) = 0 Since u1 6= 0, u1 · u1 > 0 which means c1 = : In a similar manner, c2,:::,cp can be shown to by all 0.
    [Show full text]
  • MATH 304 Linear Algebra Lecture 24: Scalar Product. Vectors: Geometric Approach
    MATH 304 Linear Algebra Lecture 24: Scalar product. Vectors: geometric approach B A B′ A′ A vector is represented by a directed segment. • Directed segment is drawn as an arrow. • Different arrows represent the same vector if • they are of the same length and direction. Vectors: geometric approach v B A v B′ A′ −→AB denotes the vector represented by the arrow with tip at B and tail at A. −→AA is called the zero vector and denoted 0. Vectors: geometric approach v B − A v B′ A′ If v = −→AB then −→BA is called the negative vector of v and denoted v. − Vector addition Given vectors a and b, their sum a + b is defined by the rule −→AB + −→BC = −→AC. That is, choose points A, B, C so that −→AB = a and −→BC = b. Then a + b = −→AC. B b a C a b + B′ b A a C ′ a + b A′ The difference of the two vectors is defined as a b = a + ( b). − − b a b − a Properties of vector addition: a + b = b + a (commutative law) (a + b) + c = a + (b + c) (associative law) a + 0 = 0 + a = a a + ( a) = ( a) + a = 0 − − Let −→AB = a. Then a + 0 = −→AB + −→BB = −→AB = a, a + ( a) = −→AB + −→BA = −→AA = 0. − Let −→AB = a, −→BC = b, and −→CD = c. Then (a + b) + c = (−→AB + −→BC) + −→CD = −→AC + −→CD = −→AD, a + (b + c) = −→AB + (−→BC + −→CD) = −→AB + −→BD = −→AD. Parallelogram rule Let −→AB = a, −→BC = b, −−→AB′ = b, and −−→B′C ′ = a. Then a + b = −→AC, b + a = −−→AC ′.
    [Show full text]
  • Math 22 – Linear Algebra and Its Applications
    Math 22 – Linear Algebra and its applications - Lecture 25 - Instructor: Bjoern Muetzel GENERAL INFORMATION ▪ Office hours: Tu 1-3 pm, Th, Sun 2-4 pm in KH 229 Tutorial: Tu, Th, Sun 7-9 pm in KH 105 ▪ Homework 8: due Wednesday at 4 pm outside KH 008. There is only Section B,C and D. 5 Eigenvalues and Eigenvectors 5.1 EIGENVECTORS AND EIGENVALUES Summary: Given a linear transformation 푇: ℝ푛 → ℝ푛, then there is always a good basis on which the transformation has a very simple form. To find this basis we have to find the eigenvalues of T. GEOMETRIC INTERPRETATION 5 −3 1 1 Example: Let 퐴 = and let 푢 = 푥 = and 푣 = . −6 2 0 2 −1 1.) Find A푣 and Au. Draw a picture of 푣 and A푣 and 푢 and A푢. 2.) Find A(3푢 +2푣) and 퐴2 (3푢 +2푣). Hint: Use part 1.) EIGENVECTORS AND EIGENVALUES ▪ Definition: An eigenvector of an 푛 × 푛 matrix A is a nonzero vector x such that 퐴푥 = 휆푥 for some scalar λ in ℝ. In this case λ is called an eigenvalue and the solution x≠ ퟎ is called an eigenvector corresponding to λ. ▪ Definition: Let A be an 푛 × 푛 matrix. The set of solutions 푛 Eig(A, λ) = {x in ℝ , such that (퐴 − 휆퐼푛)x = 0} is called the eigenspace Eig(A, λ) of A corresponding to λ. It is the null space of the matrix 퐴 − 휆퐼푛: Eig(A, λ) = Nul(퐴 − 휆퐼푛) Slide 5.1- 7 EIGENVECTORS AND EIGENVALUES 16 Example: Show that 휆 =7 is an eigenvalue of matrix A = 52 and find the corresponding eigenspace Eig(A,7).
    [Show full text]
  • Introduction to Clifford's Geometric Algebra
    解説：特集 コンピューテーショナル・インテリジェンスの新展開 —クリフォード代数表現など高次元表現を中心として— Introduction to Clifford’s Geometric Algebra Eckhard HITZER＊ ＊Department of Applied Physics, University of Fukui, Fukui, Japan ＊E-mail: [email protected] Key Words：hypercomplex algebra, hypercomplex analysis, geometry, science, engineering. JL 0004/12/5104–0338 C 2012 SICE erty 15), 21) that any isometry4 from the vector space V Abstract into an inner-product algebra5 A over the field6 K can Geometric algebra was initiated by W.K. Clifford over be uniquely extended to an isometry7 from the Clifford 130 years ago. It unifies all branches of physics, and algebra Cl(V )intoA. The Clifford algebra Cl(V )is has found rich applications in robotics, signal process- the unique associative and multilinear algebra with this ing, ray tracing, virtual reality, computer vision, vec- property. Thus if we wish to generalize methods from tor field processing, tracking, geographic information algebra, analysis, calculus, differential geometry (etc.) systems and neural computing. This tutorial explains of real numbers, complex numbers, and quaternion al- the basics of geometric algebra, with concrete exam- gebra to vector spaces and multivector spaces (which ples of the plane, of 3D space, of spacetime, and the include additional elements representing 2D up to nD popular conformal model. Geometric algebras are ideal subspaces, i.e. plane elements up to hypervolume ele- to represent geometric transformations in the general ments), the study of Clifford algebras becomes unavoid- framework of Clifford groups (also called versor or Lip- able. Indeed repeatedly and independently a long list of schitz groups). Geometric (algebra based) calculus al- Clifford algebras, their subalgebras and in Clifford al- lows e.g., to optimize learning algorithms of Clifford gebras embedded algebras (like octonions 17))ofmany neurons, etc.
    [Show full text]
  • A Bit About Hilbert Spaces
    A Bit About Hilbert Spaces David Rosenberg New York University October 29, 2016 David Rosenberg (New York University ) DS-GA 1003 October 29, 2016 1 / 9 Inner Product Space (or “Pre-Hilbert” Spaces) An inner product space (over reals) is a vector space V and an inner product, which is a mapping h·,·i : V × V ! R that has the following properties 8x,y,z 2 V and a,b 2 R: Symmetry: hx,yi = hy,xi Linearity: hax + by,zi = ahx,zi + b hy,zi Postive-definiteness: hx,xi > 0 and hx,xi = 0 () x = 0. David Rosenberg (New York University ) DS-GA 1003 October 29, 2016 2 / 9 Norm from Inner Product For an inner product space, we define a norm as p kxk = hx,xi. Example Rd with standard Euclidean inner product is an inner product space: hx,yi := xT y 8x,y 2 Rd . Norm is p kxk = xT y. David Rosenberg (New York University ) DS-GA 1003 October 29, 2016 3 / 9 What norms can we get from an inner product? Theorem (Parallelogram Law) A norm kvk can be generated by an inner product on V iff 8x,y 2 V 2kxk2 + 2kyk2 = kx + yk2 + kx - yk2, and if it can, the inner product is given by the polarization identity jjxjj2 + jjyjj2 - jjx - yjj2 hx,yi = . 2 Example d `1 norm on R is NOT generated by an inner product. [Exercise] d Is `2 norm on R generated by an inner product? David Rosenberg (New York University ) DS-GA 1003 October 29, 2016 4 / 9 Pythagorean Theroem Definition Two vectors are orthogonal if hx,yi = 0.
    [Show full text]
  • True False Questions from 4,5,6
    (c)2015 UM Math Dept licensed under a Creative Commons By-NC-SA 4.0 International License. Math 217: True False Practice Professor Karen Smith 1. For every 2 × 2 matrix A, there exists a 2 × 2 matrix B such that det(A + B) 6= det A + det B. Solution note: False! Take A to be the zero matrix for a counterexample. 2. If the change of basis matrix SA!B = ~e4 ~e3 ~e2 ~e1 , then the elements of A are the same as the element of B, but in a different order. Solution note: True. The matrix tells us that the first element of A is the fourth element of B, the second element of basis A is the third element of B, the third element of basis A is the second element of B, and the the fourth element of basis A is the first element of B. 6 6 3. Every linear transformation R ! R is an isomorphism. Solution note: False. The zero map is a a counter example. 4. If matrix B is obtained from A by swapping two rows and det A < det B, then A is invertible. Solution note: True: we only need to know det A 6= 0. But if det A = 0, then det B = − det A = 0, so det A < det B does not hold. 5. If two n × n matrices have the same determinant, they are similar. Solution note: False: The only matrix similar to the zero matrix is itself. But many 1 0 matrices besides the zero matrix have determinant zero, such as : 0 0 6.
    [Show full text]
  • Different Forms of Linear Systems, Linear Combinations, and Span
    Math 20F, 2015SS1 / TA: Jor-el Briones / Sec: A01 / Handout 2 Page 1 of2 Different forms of Linear Systems, Linear Combinations, and Span (1.3-1.4) Terms you should know Linear combination (and weights): A vector y is called a linear combination of vectors v1; v2; :::; vk if given some numbers c1; c2; :::; ck, y = c1v1 + c2v2 + ::: + ckvk The numbers c1; c2; :::; ck are called weights. Span: We call the set of ALL the possible linear combinations of a set of vectors to be the span of those vectors. For example, the span of v1 and v2 is written as span(v1; v2) NOTE: The zero vector is in the span of ANY set of vectors in Rn. Rn: The set of all vectors with n entries 3 ways to write a linear system with m equations and n unknowns If A is the m×n coefficient matrix corresponding to a linear system, with columns a1; a2; :::; an, and b in Rm is the constant vector corresponding to that linear system, we may represent the linear system using 1. A matrix equation: Ax = b 2. A vector equation: x1a1 + x2a2 + ::: + xnan = b, where x1; x2; :::xn are numbers. h i 3. An augmented matrix: A j b NOTE: You should know how to multiply a matrix by a column vector, and that doing so would result in some linear combination of the columns of that matrix. Math 20F, 2015SS1 / TA: Jor-el Briones / Sec: A01 / Handout 2 Page 2 of2 Important theorems to know: Theorem. (Chapter 1, Theorem 3) If A is an m × n matrix, with columns a1; a2; :::; an and b is in Rm, the matrix equation Ax = b has the same solution set as the vector equation x1a1 + x2a2 + ::: + xnan = b as well as the system of linear equations whose augmented matrix is h i A j b Theorem.
    [Show full text]