8. Kernel, Image, Dimension 1
|
|
- Wesley Cain
- 7 years ago
- Views:
Transcription
1 8. Kernel, Image, Dimension 1 Kernel and Image The subspace of V consisting of the vectors that are mapped to 0 in W, namely ker(t = {X V T(X = 0}, is called the kernel of the transformation T. We shall discover as we continue to flesh out the theory that, with respect to any linear transformation T:V W, the most important associated objects are the kernel of T, a subspace of V, and the image of T, Im(T = {T(X W X V }, which we have seen is a subspace of W. Examples: Consider the map R:R 3 R 3 given by the formula R(x, y, z = (x, y, z. Geometrically, this produces the effect of reflecting points in space across the xyplane. The kernel of R consists of only the zero vector: ker( R = {0}. The image is all of 3-space: Im(R = R 3.
2 8. Kernel, Image, Dimension 2 The differential operator D: P 5 (R P 5 (R can be composed with itself. The composite, D 2 = D D, is the second-order differential operator: it computes the second derivative of a polynomial in P 5 (R. Notice that ker( D 2 is the subspace P 1 (R of P 5 (R, whereas Im(D 2 is the subspace P 3 (R. Recall that, where a is any element chosen from a set S, the evaluation map e a :Fun(S,R R given by the formula e a ( f = f (a is a linear transformation. Here, ker(e a = {f :S R f (a = 0} is the subset of Fun(S,R consisting of those functions that send a to 0; that is, ker(e a = Fun(S,R;{a}. Further, the characteristic function χ a is an element of Fun(S,R, whence so is any scalar multiple, r χ a. This implies that Im( e a = R, since e a (r χ a = (r χ a (a = r(χ a (a = r. A most fundamental fact regarding a linear transformation defined on a (finite-dimensional vector space is the following theorem; it describes how the kernel and image of a transformation are related.
3 8. Kernel, Image, Dimension 3 Theorem Let T:V W be a linear transformation. If V is finite-dimensional, then so are Im(T and ker(t, and dim(im(t + dim(ker(t = dim V. (The dimension of the image space is sometimes called the rank of T, and the dimension of the kernel is sometimes called the nullity of T. As such, this theorem goes by the name of the Ranknullity Theorem. Proof Since V is finite-dimensional, so is its subspace ker(t, so we can find a basis for it, say {B 1, B 2,, B k }. Extend this set to a basis for all of V by including the vectors A 1, A 2,, A j ; that is, {A 1, A 2,, A j, B 1, B 2,, B k } is a basis for V. Then Im(T = T(V = T(L(A 1, A 2,, A j, B 1, B 2,, B k = L(T(A 1,, T(A j, T(B 1,, T(B k = L(T(A 1,, T(A j,0,,0 = L(T(A 1,, T(A j shows that Im(T is finite-dimensional as well.
4 8. Kernel, Image, Dimension 4 Indeed, it also shows that Im(T is spanned by the j vectors T(A 1, T(A 2,, T(A j. We claim that these j vectors are in fact linearly independent as well. For if there are scalars r 1,r 2,,r j for which r 1 T(A 1 + r 2 T(A 2 +L + r j T(A j = 0, then by the linearity of T, T(r 1 A 1 + r 2 A 2 +L + r j A j = 0, implying that r 1 A 1 + r 2 A 2 +L + r j A j lies in ker(t. Therefore, there must equal scalars s 1, s 2,, s k for which r 1 A 1 + r 2 A 2 +L + r j A j = s 1 B 1 + s 2 B 2 +L + s k B k. Writing this in the form r 1 A 1 + r 2 A 2 +L + r j A j s 1 B 1 s 2 B 2 L s k B k = 0 gives us a linear combination of basis vectors of V equal to the zero vector, so all the scalars in particular, all the r s must equal 0. Thus, we have shown that dim(im(t = j, dim(ker(t = k, and that dim V = j + k, which is what the theorem states. //
5 8. Kernel, Image, Dimension 5 Fun with linear transformations We have seen how, given an arbitrary set S and a vector space W, the function space Fun(S,W is itself a vector space under natural definitions of addition and scalar multiplication of functions from S to W. (Recall Exercise 3.10, p. 31. An important example of such a vector space arises when S is chosen to be a basis for another vector space V. Let S = {A 1, A 2,, A m } be a basis for V. Then choose some function f from Fun(S,W and set f(a 1 = B 1, f (A 2 = B 2,, f(a m = B m. These B s are vectors in W; they need not even be distinct. Now, since every A V is uniquely a linear combination of vectors from S, we can define a new function F from V to W by setting equal to F(A = F(r 1 A 1 + r 2 A 2 +L + r m A m r 1 f(a 1 + r 2 f (A 2 +L + r m f(a m = r 1 B 1 + r 2 B 2 +L + r m B m.
6 8. Kernel, Image, Dimension 6 In fact, this definition shows that F(A i = f(a i for each i = 1,2,,m; that is, F can be viewed as an extension of the function f : f has S as domain whereas F has all of V = L(S as domain. What is more, F is a linear transformation between the vector spaces V andw since F(r 1 A 1 + r 2 A 2 +L + r m A m = r 1 f(a 1 + r 2 f (A 2 +L + r m f(a m = r 1 F(A 1 + r 2 F(A 2 +L + r m F(A m For this reason, we call F the linear extension of f. It is a useful construction, as we shall now see. Theorem Let A 1, A 2,, A m V be any m vectors chosen from a vector space V. Where E = {E 1, E 2,, E m } is the standard basis in R m, define f Fun(E,V as f(e i = A i for each i and let F be its linear extension to all of R m. Then (1 {A 1, A 2,, A m } is linearly independent in V if and only if ker(f = {0}; and (2 {A 1, A 2,, A m } spans V if and only if Im(F = V.
7 8. Kernel, Image, Dimension 7 Proof (1 {A 1, A 2,, A m } is linearly independent in V whenever the linear combination r 1 A 1 + r 2 A 2 +L + r m A m equals 0, r 1 = r 2 = L = r m = 0 whenever 0 = r 1 f(e 1 + r 2 f (E 2 +L + r m f(e m = r 1 F(E 1 + r 2 F(E 2 +L + r m F(E m = F(r 1 E 1 + r 2 E 2 +L + r m E m = F(r 1,r 2,,r m, then r 1 = r 2 = L = r m = 0 if X = (r 1,r 2,,r m ker(f, then X = 0 ker(f = {0}. (2 {A 1, A 2,, A m } spans V every A V has the form r 1 A 1 + r 2 A 2 +L + r m A m = r 1 f(e 1 + r 2 f (E 2 +L + r m f(e m Im(F = V. // = r 1 F(E 1 + r 2 F(E 2 +L + r m F(E m = F(r 1 E 1 + r 2 E 2 +L + r m E m = F(r 1,r 2,,r m
8 8. Kernel, Image, Dimension 8 We know that Fun(S,W, the collection of functions f:s W, has the structure of a vector space. When we choose this set S = {A 1, A 2,, A m } to be a basis for another vector space V, then the collection of linear extensions F:V W of elements of Fun(S,W, which we denote L(V,W, is itself a vector space, in fact, a subspace of the vector space Fun(V,W. (Why? Observe that every linear transformation T:V W between V and W is the linear extension of some function in Fun(S,W, namely of that function whose values at each vector in S is the same as the value T has there. Thus, L(V,W is the space of all linear transformations between V and W. A particularly important example of a space of linear transformations arises when W is taken to be the 1-dimensional vector space consisting of the field of scalars R. In this instance, we call L(V,R the dual space of V and use the special notation: V* = L(V,R. We will study the properties of the dual space of a vector space somewhat later.
9 8. Kernel, Image, Dimension 9 Another important situation occurs when W = V. Here, L(V,V represents the space of linear transformations from V to itself. Such functions are called endomorphisms on V. The simplest endomorphism on a vector space is the identity map I L(V,V, with the simple definition I(A = A. Yet another important phenomeon arises when a linear transformation is invertible: if T L(V,W and S L(W,V, then these transformations can be composed in both directions: S o T L(V,V and T o S L(W,W are both endomorphisms (but of different spaces!. If it should happen that both S o T and T o S are identity maps on their respective spaces, then T is invertible and S is its inverse transformation. In addition, since every invertible function must be a one-to-one correspondence between its domain and codomain, every invertible transformation T L(V,W is a one-to-one correspondence between the vector spaces V and W. That is, every vector A in V is associated by the transformation T to a unique vector T(A in W, and vice versa. The linearity of T ensures, for instance, that a set of basis vectors in V is associated to a set of basis vectors in W. In particular, this means that if there is an invertible
10 8. Kernel, Image, Dimension 10 linear transformation T L(V,W, then V and W must have the same dimension, indeed, the same structure. Consequently, we call the two vector spaces isomorphic and also refer to T as a vector space isomorphism. Proposition Suppose the linear transformation T L(V,W has trivial kernel (ker(t = {0}. Then every vector B Im(T is the image of a unique vector A V. Proof Clearly, B is the image of at least one vector in V. But if both A and A have B as image, then T(A = T( A, so T(A A = T(A T( A = 0, so A A ker(t. So A A = 0, whence A = A. // Theorem A linear transformation T L(V,W is an isomorphism (i.e., is invertible if and only if ker(t = {0} and Im(T = W (the kernel is trivial and the image is full. Proof Suppose first that T L(V,W is an isomorphism. Then it has an inverse linear transformation S L(W,V for which the compositions S o T and T o S are identity maps. So if A ker(t, then A = S o T(A = S(0 = 0; that is,
11 8. Kernel, Image, Dimension 11 ker(t = {0}. Also, if B W, then B = T o S(B is the image under T of the vector S(B V; that is, Im(T = W. Conversely, suppose that T L(V,W is a linear transformation with trivial kernel and full image. Then by the previous proposition, every vector B in Im(T = W is the image of a unique vector A V. So we can define a function S:W V that sends B to the unique vector A that is sent to B by T. This function is a linear transformation, for if S(B = A and S( B = A, then T(A = B and T( A = B, so T(A + A = B + B, implying that S(B + B = A + A = S(B + S( B ; also, for any scalar r, T(rA = rb so S(rB = ra. Thus, S L(W,V and S o T and T o S are the respective identity maps, so T is an isomorphism. //
12 8. Kernel, Image, Dimension 12 Corollary If V and W are finite-dimensional vector spaces, then they are isomorphic if and only if they have the same dimension. Proof If V and W are isomorphic, then there is some isomorphsim T:V W between them. If V has dimension n, then it has a basis with n vectors, say {A 1, A 2,, A n } is a basis for V. Then W = T(V = T(L(A 1, A 2,, A n = L(T(A 1,, T(A 2,, T(A n so that {T(A 1, T(A 2,, T(A n } is a spanning set for W. But these vectors are also linearly independent, for if r 1 T(A 1 + r 2 T(A 2 +L + r n T(A n = 0, then T(r 1 A 1 + r 2 A 2 +L + r n A n = 0, which, since T has trivial kernel, means that r 1 A 1 + r 2 A 2 +L + r n A n = 0, whence, since {A 1, A 2,, A n } is a basis, forces all the r s to equal 0. Therefore,
13 8. Kernel, Image, Dimension 13 {T(A 1, T(A 2,, T(A n } is a basis for W, from which we conclude that W also has dimension n. Conversely, if dimv = dimw = n, then both spaces have bases of size n: say {A 1, A 2,, A n } is a basis for V and {B 1, B 2,, B n } is a basis for W. Let T be the linear extension of the function that sends A i to B i for i = 1, 2,, n. Then and also, Im(T = L(B 1, B 2,, B n = W, n = dim(v = dim(im(t + dim(ker(t = dim(w + dim(ker(t = n + dim(ker(t implying that dim(ker(t = 0, so ker(t = {0}. Since T has trivial kernel and full image, it must be an isomorphism. // Corollary Every n-dimensional vector space is isomorphic to R n. //
NOTES ON LINEAR TRANSFORMATIONS
NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all
More informationMA106 Linear Algebra lecture notes
MA106 Linear Algebra lecture notes Lecturers: Martin Bright and Daan Krammer Warwick, January 2011 Contents 1 Number systems and fields 3 1.1 Axioms for number systems......................... 3 2 Vector
More informationLinear Algebra. A vector space (over R) is an ordered quadruple. such that V is a set; 0 V ; and the following eight axioms hold:
Linear Algebra A vector space (over R) is an ordered quadruple (V, 0, α, µ) such that V is a set; 0 V ; and the following eight axioms hold: α : V V V and µ : R V V ; (i) α(α(u, v), w) = α(u, α(v, w)),
More informationLinear Maps. Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007)
MAT067 University of California, Davis Winter 2007 Linear Maps Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007) As we have discussed in the lecture on What is Linear Algebra? one of
More informationMATH1231 Algebra, 2015 Chapter 7: Linear maps
MATH1231 Algebra, 2015 Chapter 7: Linear maps A/Prof. Daniel Chan School of Mathematics and Statistics University of New South Wales danielc@unsw.edu.au Daniel Chan (UNSW) MATH1231 Algebra 1 / 43 Chapter
More informationT ( a i x i ) = a i T (x i ).
Chapter 2 Defn 1. (p. 65) Let V and W be vector spaces (over F ). We call a function T : V W a linear transformation form V to W if, for all x, y V and c F, we have (a) T (x + y) = T (x) + T (y) and (b)
More informationLINEAR ALGEBRA W W L CHEN
LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,
More informationMATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix.
MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix. Nullspace Let A = (a ij ) be an m n matrix. Definition. The nullspace of the matrix A, denoted N(A), is the set of all n-dimensional column
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationMatrix Representations of Linear Transformations and Changes of Coordinates
Matrix Representations of Linear Transformations and Changes of Coordinates 01 Subspaces and Bases 011 Definitions A subspace V of R n is a subset of R n that contains the zero element and is closed under
More information5. Linear algebra I: dimension
5. Linear algebra I: dimension 5.1 Some simple results 5.2 Bases and dimension 5.3 Homomorphisms and dimension 1. Some simple results Several observations should be made. Once stated explicitly, the proofs
More information160 CHAPTER 4. VECTOR SPACES
160 CHAPTER 4. VECTOR SPACES 4. Rank and Nullity In this section, we look at relationships between the row space, column space, null space of a matrix and its transpose. We will derive fundamental results
More informationMATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.
MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V V (x,y) x + y V and scalar
More informationOrthogonal Diagonalization of Symmetric Matrices
MATH10212 Linear Algebra Brief lecture notes 57 Gram Schmidt Process enables us to find an orthogonal basis of a subspace. Let u 1,..., u k be a basis of a subspace V of R n. We begin the process of finding
More informationSolutions to Math 51 First Exam January 29, 2015
Solutions to Math 5 First Exam January 29, 25. ( points) (a) Complete the following sentence: A set of vectors {v,..., v k } is defined to be linearly dependent if (2 points) there exist c,... c k R, not
More informationMath 333 - Practice Exam 2 with Some Solutions
Math 333 - Practice Exam 2 with Some Solutions (Note that the exam will NOT be this long) Definitions (0 points) Let T : V W be a transformation Let A be a square matrix (a) Define T is linear (b) Define
More informationSimilarity and Diagonalization. Similar Matrices
MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that
More information4.5 Linear Dependence and Linear Independence
4.5 Linear Dependence and Linear Independence 267 32. {v 1, v 2 }, where v 1, v 2 are collinear vectors in R 3. 33. Prove that if S and S are subsets of a vector space V such that S is a subset of S, then
More information1 Sets and Set Notation.
LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most
More informationTHE DIMENSION OF A VECTOR SPACE
THE DIMENSION OF A VECTOR SPACE KEITH CONRAD This handout is a supplementary discussion leading up to the definition of dimension and some of its basic properties. Let V be a vector space over a field
More informationUniversity of Lille I PC first year list of exercises n 7. Review
University of Lille I PC first year list of exercises n 7 Review Exercise Solve the following systems in 4 different ways (by substitution, by the Gauss method, by inverting the matrix of coefficients
More information16.3 Fredholm Operators
Lectures 16 and 17 16.3 Fredholm Operators A nice way to think about compact operators is to show that set of compact operators is the closure of the set of finite rank operator in operator norm. In this
More information1 0 5 3 3 A = 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0
Solutions: Assignment 4.. Find the redundant column vectors of the given matrix A by inspection. Then find a basis of the image of A and a basis of the kernel of A. 5 A The second and third columns are
More informationSection 6.1 - Inner Products and Norms
Section 6.1 - Inner Products and Norms Definition. Let V be a vector space over F {R, C}. An inner product on V is a function that assigns, to every ordered pair of vectors x and y in V, a scalar in F,
More informationI. GROUPS: BASIC DEFINITIONS AND EXAMPLES
I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called
More informationLEARNING OBJECTIVES FOR THIS CHAPTER
CHAPTER 2 American mathematician Paul Halmos (1916 2006), who in 1942 published the first modern linear algebra book. The title of Halmos s book was the same as the title of this chapter. Finite-Dimensional
More informationFinite dimensional topological vector spaces
Chapter 3 Finite dimensional topological vector spaces 3.1 Finite dimensional Hausdorff t.v.s. Let X be a vector space over the field K of real or complex numbers. We know from linear algebra that the
More informationLinear Algebra Notes
Linear Algebra Notes Chapter 19 KERNEL AND IMAGE OF A MATRIX Take an n m matrix a 11 a 12 a 1m a 21 a 22 a 2m a n1 a n2 a nm and think of it as a function A : R m R n The kernel of A is defined as Note
More informationName: Section Registered In:
Name: Section Registered In: Math 125 Exam 3 Version 1 April 24, 2006 60 total points possible 1. (5pts) Use Cramer s Rule to solve 3x + 4y = 30 x 2y = 8. Be sure to show enough detail that shows you are
More informationby the matrix A results in a vector which is a reflection of the given
Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the y-axis We observe that
More informationMATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.
MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column
More informationBANACH AND HILBERT SPACE REVIEW
BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but
More informationSection 1.7 22 Continued
Section 1.5 23 A homogeneous equation is always consistent. TRUE - The trivial solution is always a solution. The equation Ax = 0 gives an explicit descriptions of its solution set. FALSE - The equation
More informationLinear Algebra I. Ronald van Luijk, 2012
Linear Algebra I Ronald van Luijk, 2012 With many parts from Linear Algebra I by Michael Stoll, 2007 Contents 1. Vector spaces 3 1.1. Examples 3 1.2. Fields 4 1.3. The field of complex numbers. 6 1.4.
More information1 VECTOR SPACES AND SUBSPACES
1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such
More information( ) which must be a vector
MATH 37 Linear Transformations from Rn to Rm Dr. Neal, WKU Let T : R n R m be a function which maps vectors from R n to R m. Then T is called a linear transformation if the following two properties are
More informationChapter 19. General Matrices. An n m matrix is an array. a 11 a 12 a 1m a 21 a 22 a 2m A = a n1 a n2 a nm. The matrix A has n row vectors
Chapter 9. General Matrices An n m matrix is an array a a a m a a a m... = [a ij]. a n a n a nm The matrix A has n row vectors and m column vectors row i (A) = [a i, a i,..., a im ] R m a j a j a nj col
More information17. Inner product spaces Definition 17.1. Let V be a real vector space. An inner product on V is a function
17. Inner product spaces Definition 17.1. Let V be a real vector space. An inner product on V is a function, : V V R, which is symmetric, that is u, v = v, u. bilinear, that is linear (in both factors):
More informationRow Ideals and Fibers of Morphisms
Michigan Math. J. 57 (2008) Row Ideals and Fibers of Morphisms David Eisenbud & Bernd Ulrich Affectionately dedicated to Mel Hochster, who has been an inspiration to us for many years, on the occasion
More informationPolynomial Invariants
Polynomial Invariants Dylan Wilson October 9, 2014 (1) Today we will be interested in the following Question 1.1. What are all the possible polynomials in two variables f(x, y) such that f(x, y) = f(y,
More informationThe cover SU(2) SO(3) and related topics
The cover SU(2) SO(3) and related topics Iordan Ganev December 2011 Abstract The subgroup U of unit quaternions is isomorphic to SU(2) and is a double cover of SO(3). This allows a simple computation of
More informationPROJECTIVE GEOMETRY. b3 course 2003. Nigel Hitchin
PROJECTIVE GEOMETRY b3 course 2003 Nigel Hitchin hitchin@maths.ox.ac.uk 1 1 Introduction This is a course on projective geometry. Probably your idea of geometry in the past has been based on triangles
More informationSystems of Linear Equations
Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and
More informationChapter 20. Vector Spaces and Bases
Chapter 20. Vector Spaces and Bases In this course, we have proceeded step-by-step through low-dimensional Linear Algebra. We have looked at lines, planes, hyperplanes, and have seen that there is no limit
More informationα = u v. In other words, Orthogonal Projection
Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v
More information3. Prime and maximal ideals. 3.1. Definitions and Examples.
COMMUTATIVE ALGEBRA 5 3.1. Definitions and Examples. 3. Prime and maximal ideals Definition. An ideal P in a ring A is called prime if P A and if for every pair x, y of elements in A\P we have xy P. Equivalently,
More informationGROUP ALGEBRAS. ANDREI YAFAEV
GROUP ALGEBRAS. ANDREI YAFAEV We will associate a certain algebra to a finite group and prove that it is semisimple. Then we will apply Wedderburn s theory to its study. Definition 0.1. Let G be a finite
More informationScalar Valued Functions of Several Variables; the Gradient Vector
Scalar Valued Functions of Several Variables; the Gradient Vector Scalar Valued Functions vector valued function of n variables: Let us consider a scalar (i.e., numerical, rather than y = φ(x = φ(x 1,
More information4.1 Modules, Homomorphisms, and Exact Sequences
Chapter 4 Modules We always assume that R is a ring with unity 1 R. 4.1 Modules, Homomorphisms, and Exact Sequences A fundamental example of groups is the symmetric group S Ω on a set Ω. By Cayley s Theorem,
More informationVector Spaces 4.4 Spanning and Independence
Vector Spaces 4.4 and Independence October 18 Goals Discuss two important basic concepts: Define linear combination of vectors. Define Span(S) of a set S of vectors. Define linear Independence of a set
More informationInner Product Spaces
Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and
More informationFINITE DIMENSIONAL ORDERED VECTOR SPACES WITH RIESZ INTERPOLATION AND EFFROS-SHEN S UNIMODULARITY CONJECTURE AARON TIKUISIS
FINITE DIMENSIONAL ORDERED VECTOR SPACES WITH RIESZ INTERPOLATION AND EFFROS-SHEN S UNIMODULARITY CONJECTURE AARON TIKUISIS Abstract. It is shown that, for any field F R, any ordered vector space structure
More informationRecall that two vectors in are perpendicular or orthogonal provided that their dot
Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal
More information4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION
4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION STEVEN HEILMAN Contents 1. Review 1 2. Diagonal Matrices 1 3. Eigenvectors and Eigenvalues 2 4. Characteristic Polynomial 4 5. Diagonalizability 6 6. Appendix:
More informationSection 4.4 Inner Product Spaces
Section 4.4 Inner Product Spaces In our discussion of vector spaces the specific nature of F as a field, other than the fact that it is a field, has played virtually no role. In this section we no longer
More informationChapter 6. Linear Transformation. 6.1 Intro. to Linear Transformation
Chapter 6 Linear Transformation 6 Intro to Linear Transformation Homework: Textbook, 6 Ex, 5, 9,, 5,, 7, 9,5, 55, 57, 6(a,b), 6; page 7- In this section, we discuss linear transformations 89 9 CHAPTER
More informationLS.6 Solution Matrices
LS.6 Solution Matrices In the literature, solutions to linear systems often are expressed using square matrices rather than vectors. You need to get used to the terminology. As before, we state the definitions
More informationOn the representability of the bi-uniform matroid
On the representability of the bi-uniform matroid Simeon Ball, Carles Padró, Zsuzsa Weiner and Chaoping Xing August 3, 2012 Abstract Every bi-uniform matroid is representable over all sufficiently large
More informationa 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given
More informationMATH 551 - APPLIED MATRIX THEORY
MATH 55 - APPLIED MATRIX THEORY FINAL TEST: SAMPLE with SOLUTIONS (25 points NAME: PROBLEM (3 points A web of 5 pages is described by a directed graph whose matrix is given by A Do the following ( points
More informationMath 67: Modern Linear Algebra (UC Davis, Fall 2011) Summary of lectures Prof. Dan Romik
Math 67: Modern Linear Algebra (UC Davis, Fall 2011) Summary of lectures Prof. Dan Romik [Version of November 30, 2011 this document will be updated occasionally with new material] Lecture 1 (9/23/11)
More informationLinearly Independent Sets and Linearly Dependent Sets
These notes closely follow the presentation of the material given in David C. Lay s textbook Linear Algebra and its Applications (3rd edition). These notes are intended primarily for in-class presentation
More informationAu = = = 3u. Aw = = = 2w. so the action of A on u and w is very easy to picture: it simply amounts to a stretching by 3 and 2, respectively.
Chapter 7 Eigenvalues and Eigenvectors In this last chapter of our exploration of Linear Algebra we will revisit eigenvalues and eigenvectors of matrices, concepts that were already introduced in Geometry
More information88 CHAPTER 2. VECTOR FUNCTIONS. . First, we need to compute T (s). a By definition, r (s) T (s) = 1 a sin s a. sin s a, cos s a
88 CHAPTER. VECTOR FUNCTIONS.4 Curvature.4.1 Definitions and Examples The notion of curvature measures how sharply a curve bends. We would expect the curvature to be 0 for a straight line, to be very small
More informationLinear Equations in Linear Algebra
1 Linear Equations in Linear Algebra 1.5 SOLUTION SETS OF LINEAR SYSTEMS HOMOGENEOUS LINEAR SYSTEMS A system of linear equations is said to be homogeneous if it can be written in the form A 0, where A
More informationOrthogonal Projections and Orthonormal Bases
CS 3, HANDOUT -A, 3 November 04 (adjusted on 7 November 04) Orthogonal Projections and Orthonormal Bases (continuation of Handout 07 of 6 September 04) Definition (Orthogonality, length, unit vectors).
More information1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or
More informationLinear Algebra Notes for Marsden and Tromba Vector Calculus
Linear Algebra Notes for Marsden and Tromba Vector Calculus n-dimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of
More informationHow To Prove The Dirichlet Unit Theorem
Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if
More informationChapter 13: Basic ring theory
Chapter 3: Basic ring theory Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 42, Spring 24 M. Macauley (Clemson) Chapter 3: Basic ring
More informationChapter 17. Orthogonal Matrices and Symmetries of Space
Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length
More information1 Introduction to Matrices
1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns
More informationNotes on Symmetric Matrices
CPSC 536N: Randomized Algorithms 2011-12 Term 2 Notes on Symmetric Matrices Prof. Nick Harvey University of British Columbia 1 Symmetric Matrices We review some basic results concerning symmetric matrices.
More informationIRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction
IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible
More informationFUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES
FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied
More informationMethods for Finding Bases
Methods for Finding Bases Bases for the subspaces of a matrix Row-reduction methods can be used to find bases. Let us now look at an example illustrating how to obtain bases for the row space, null space,
More informationISOMETRIES OF R n KEITH CONRAD
ISOMETRIES OF R n KEITH CONRAD 1. Introduction An isometry of R n is a function h: R n R n that preserves the distance between vectors: h(v) h(w) = v w for all v and w in R n, where (x 1,..., x n ) = x
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
More informationLinear Algebra Review. Vectors
Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka kosecka@cs.gmu.edu http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa Cogsci 8F Linear Algebra review UCSD Vectors The length
More informationArkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 3 Binary Operations We are used to addition and multiplication of real numbers. These operations combine two real numbers
More information8.1 Examples, definitions, and basic properties
8 De Rham cohomology Last updated: May 21, 211. 8.1 Examples, definitions, and basic properties A k-form ω Ω k (M) is closed if dω =. It is exact if there is a (k 1)-form σ Ω k 1 (M) such that dσ = ω.
More informationIdeal Class Group and Units
Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals
More informationLinear Algebra Done Wrong. Sergei Treil. Department of Mathematics, Brown University
Linear Algebra Done Wrong Sergei Treil Department of Mathematics, Brown University Copyright c Sergei Treil, 2004, 2009, 2011, 2014 Preface The title of the book sounds a bit mysterious. Why should anyone
More information5 Homogeneous systems
5 Homogeneous systems Definition: A homogeneous (ho-mo-jeen -i-us) system of linear algebraic equations is one in which all the numbers on the right hand side are equal to : a x +... + a n x n =.. a m
More informationLecture 18 - Clifford Algebras and Spin groups
Lecture 18 - Clifford Algebras and Spin groups April 5, 2013 Reference: Lawson and Michelsohn, Spin Geometry. 1 Universal Property If V is a vector space over R or C, let q be any quadratic form, meaning
More informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J. Olver 6. Eigenvalues and Singular Values In this section, we collect together the basic facts about eigenvalues and eigenvectors. From a geometrical viewpoint,
More informationQuotient Rings and Field Extensions
Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.
More informationInner Product Spaces and Orthogonality
Inner Product Spaces and Orthogonality week 3-4 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,
More informationNotes on Determinant
ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without
More informationThese axioms must hold for all vectors ū, v, and w in V and all scalars c and d.
DEFINITION: A vector space is a nonempty set V of objects, called vectors, on which are defined two operations, called addition and multiplication by scalars (real numbers), subject to the following axioms
More informationLinear Algebra Done Wrong. Sergei Treil. Department of Mathematics, Brown University
Linear Algebra Done Wrong Sergei Treil Department of Mathematics, Brown University Copyright c Sergei Treil, 2004, 2009, 2011, 2014 Preface The title of the book sounds a bit mysterious. Why should anyone
More informationNOTES ON CATEGORIES AND FUNCTORS
NOTES ON CATEGORIES AND FUNCTORS These notes collect basic definitions and facts about categories and functors that have been mentioned in the Homological Algebra course. For further reading about category
More informationMath 4310 Handout - Quotient Vector Spaces
Math 4310 Handout - Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable
More informationChapter 5. Banach Spaces
9 Chapter 5 Banach Spaces Many linear equations may be formulated in terms of a suitable linear operator acting on a Banach space. In this chapter, we study Banach spaces and linear operators acting on
More informationA Modern Course on Curves and Surfaces. Richard S. Palais
A Modern Course on Curves and Surfaces Richard S. Palais Contents Lecture 1. Introduction 1 Lecture 2. What is Geometry 4 Lecture 3. Geometry of Inner-Product Spaces 7 Lecture 4. Linear Maps and the Euclidean
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More informationSMALL SKEW FIELDS CÉDRIC MILLIET
SMALL SKEW FIELDS CÉDRIC MILLIET Abstract A division ring of positive characteristic with countably many pure types is a field Wedderburn showed in 1905 that finite fields are commutative As for infinite
More informationCartesian Products and Relations
Cartesian Products and Relations Definition (Cartesian product) If A and B are sets, the Cartesian product of A and B is the set A B = {(a, b) :(a A) and (b B)}. The following points are worth special
More informationJim Lambers MAT 169 Fall Semester 2009-10 Lecture 25 Notes
Jim Lambers MAT 169 Fall Semester 009-10 Lecture 5 Notes These notes correspond to Section 10.5 in the text. Equations of Lines A line can be viewed, conceptually, as the set of all points in space that
More informationSection 1.4. Lines, Planes, and Hyperplanes. The Calculus of Functions of Several Variables
The Calculus of Functions of Several Variables Section 1.4 Lines, Planes, Hyperplanes In this section we will add to our basic geometric understing of R n by studying lines planes. If we do this carefully,
More information