# 5. Linear algebra I: dimension

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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 are easy. [1] The intersection of a (non-empty) set of subspaces of a vector space V is a subspace. Proof: Let {W i : i I} be a set of subspaces of V. For w in every W i, the additive inverse w is in W i. Thus, w lies in the intersection. The same argument proves the other properties of subspaces. The subspace spanned by a set X of vectors in a vector space V is the intersection of all subspaces containing X. From above, this intersection is a subspace. The subspace spanned by a set X in a vector space V is the collection of all linear combinations of vectors from X. Proof: Certainly every linear combination of vectors taken from X is in any subspace containing X. On the other hand, we must show that any vector in the intersection of subspaces containing X is a linear combination of vectors in X. Now it is not hard to check that the collection of such linear combinations is itself a subspace of V, and contains X. Therefore, the intersection is no larger than this set of linear combinations. [1] At the beginning of the abstract form of this and other topics, there are several results which have little informational content, but, rather, only serve to assure us that the definitions/axioms have not included phenomena too violently in opposition to our expectations. This is not surprising, considering that the definitions have endured several decades of revision exactly to address foundational and other potential problems. 85

2 86 Linear algebra I: dimension A linearly independent set of vectors spanning a subspace W of V is a basis for W. [1.0.1] Proposition: Given a basis e 1,..., e n for a vector space V, there is exactly one expression for an arbitrary vector v V as a linear combination of e 1,..., e n. Proof: That there is at least one expression follows from the spanning property. On the other hand, if are two expressions for v, then subtract to obtain a i e i = v = i i (a i b i )e i = 0 i Since the e i are linearly independent, a i = b i for all indices i. b i e i 2. Bases and dimension The argument in the proof of the following fundamental theorem is the Lagrange replacement principle. This is the first non-trivial result in linear algebra. [2.0.1] Theorem: Let v 1,..., v m be a linearly independent set of vectors in a vector space V, and let w 1,..., w n be a basis for V. Then m n, and (renumbering the vectors w i if necessary) the vectors are a basis for V. v 1,..., v m, w m+1, w m+2,..., w n Proof: Since the w i s are a basis, we may express v 1 as a linear combination v 1 = c 1 w c n w n Not all coefficients can be 0, since v 1 is not 0. Renumbering the w i s if necessary, we can assume that c 1 0. Since the scalars k are a field, we can express w 1 in terms of v 1 and w 2,..., w n w 1 = c 1 1 v 1 + ( c 1 1 c 2)w ( c 1 1 c 2)w n Replacing w 1 by v 1, the vectors v 1, w 2, w 3,..., w n span V. They are still linearly independent, since if v 1 were a linear combination of w 2,..., w n then the expression for w 1 in terms of v 1, w 2,..., w n would show that w 1 was a linear combination of w 2,..., w n, contradicting the linear independence of w 1,..., w n. Suppose inductively that v 1,..., v i, w i+1,..., w n are a basis for V, with i < n. Express v i+1 as a linear combination v i+1 = a 1 v a i v i + b i+1 w i b n w n Some b j is non-zero, or else v i is a linear combination of v 1,..., v i, contradicting the linear independence of the v j s. By renumbering the w j s if necessary, assume that b i+1 0. Rewrite this to express w i+1 as a linear combination of v 1,..., v i, w i+1,..., w n w i+1 = ( b 1 i+1 a 1)v ( b 1 i+1 a i)v i + (b 1 i+1 )v i+1 + ( b 1 i+1 b i+2)w i ( b 1 i+1 b n)w n

3 Garrett: Abstract Algebra 87 Thus, v 1,..., v i+1, w i+2,..., w n span V. Claim that these vectors are linearly independent: if for some coefficients a j, b j a 1 v a i+1 v i+1 + b i+2 w i b n w n = 0 then some a i+1 is non-zero, because of the linear independence of v 1,..., v i, w i+1,..., w n. Thus, rearrange to express v i+1 as a linear combination of v 1,..., v i, w i+2,..., w n. The expression for w i+1 in terms of v 1,..., v i, v i+1, w i+2,..., w n becomes an expression for w i+1 as a linear combination of v 1,..., v i, w i+2,..., w n. But this would contradict the (inductively assumed) linear independence of v 1,..., v i, w i+1, w i+2,..., w n. Consider the possibility that m > n. Then, by the previous argument, v 1,..., v n is a basis for V. Thus, v n+1 is a linear combination of v 1,..., v n, contradicting their linear independence. Thus, m n, and v 1,..., v m, w m+1,..., w n is a basis for V, as claimed. Now define the (k-)dimension [2] of a vector space (over field k) as the number of elements in a (k-)basis. The theorem says that this number is well-defined. Write dim V = dimension of V A vector space is finite-dimensional if it has a finite basis. [3] [2.0.2] Corollary: A linearly independent set of vectors in a finite-dimensional vector space can be augmented to be a basis. Proof: Let v 1,..., v m be as linearly independent set of vectors, let w 1,..., w n be a basis, and apply the theorem. [2.0.3] Corollary: The dimension of a proper subspace of a finite-dimensional vector space is strictly less than the dimension of the whole space. Proof: Let w 1,..., w m be a basis for the subspace. By the theorem, it can be extended to a basis w 1,..., w m, v m+1,..., v n of the whole space. It must be that n > m, or else the subspace is the whole space. [2.0.4] Corollary: The dimension of k n is n. The vectors are a basis (the standard basis). Proof: Those vectors span k n, since e 1 = (1, 0, 0,..., 0, 0) e 2 = (0, 1, 0,..., 0, 0) e 3 = (0, 0, 1,..., 0, 0)... e n = (0, 0, 0,..., 0, 1) (c 1,..., c n ) = c 1 e c n e n [2] This is an instance of terminology that is nearly too suggestive. That is, a naive person might all too easily accidentally assume that there is a connection to the colloquial sense of the word dimension, or that there is an appeal to physical or visual intuition. Or one might assume that it is somehow obvious that dimension is a welldefined invariant. [3] We proved only the finite-dimensional case of the well-definedness of dimension. The infinite-dimensional case needs transfinite induction or an equivalent.

4 88 Linear algebra I: dimension On the other hand, a linear dependence relation gives 0 = c 1 e c n e n (c 1,..., c n ) = (0,..., 0) from which each c i is 0. Thus, these vectors are a basis for k n. 3. Homomorphisms and dimension Now we see how dimension behaves under homomorphisms. Again, a vector space homomorphism [4] f : V W from a vector space V over a field k to a vector space W over the same field k is a function f such that The kernel of f is and the image of f is f(v 1 + v 2 ) = f(v 1 ) + f(v 2 ) (for all v 1, v 2 V ) f(α v) = α f(v) (for all α k, v V ) ker f = {v V : f(v) = 0} Imf = {f(v) : v V } A homomorphism is an isomorphism if it has a two-sided inverse homomorphism. For vector spaces, a homomorphism that is a bijection is an isomorphism. [5] A vector space homomorphism f : V W sends 0 (in V ) to 0 (in W, and, for v V, f( v) = f(v). [6] [3.0.1] Proposition: The kernel and image of a vector space homomorphism f : V W are vector subspaces of V and W, respectively. Proof: Regarding the kernel, the previous proposition shows that it contains 0. The last bulleted point was that additive inverses of elements in the kernel are again in the kernel. For x, y ker f f(x + y) = f(x) + f(y) = = 0 so ker f is closed under addition. For α k and v V f(α v) = α f(v) = α 0 = 0 so ker f is closed under scalar multiplication. Thus, the kernel is a vector subspace. Similarly, f(0) = 0 shows that 0 is in the image of f. For w = f(v) in the image of f and α k α w = α f(v) = f(αv) Imf [4] Or linear map or linear operator. [5] In most of the situations we will encounter, bijectivity of various sorts of homomorphisms is sufficient (and certainly necessary) to assure that there is an inverse map of the same sort, justifying this description of isomorphism. [6] This follows from the analogous result for groups, since V with its additive structure is an abelian group.

5 For x = f(u) and y = f(v) both in the image of f, Garrett: Abstract Algebra 89 x + y = f(u) + f(v) = f(u + v) Imf And from above so the image is a vector subspace. f( v) = f(v) [3.0.2] Corollary: A linear map f : V W is injective if and only if its kernel is the trivial subspace {0}. Proof: This follows from the analogous assertion for groups. [3.0.3] Corollary: Let f : V W be a vector space homomorphism, with V finite-dimensional. Then dim ker f + dim Imf = dim V Proof: Let v 1,..., v m be a basis for ker f, and, invoking the theorem, let w m+1,..., w n be vectors in V such that v 1,..., v m, w m+1,..., w n form a basis for V. We claim that the images f(w m+1 ),..., f(w n ) are a basis for Imf. First, show that these vectors span. For f(v) = w, express v as a linear combination and apply f v = a 1 v a m v m + b m+1 w m b n w n w = a 1 f(v 1 ) a m f(v m ) + b m+1 f(w m+1 ) b n f(w n ) = a a m 0(v m ) + b m+1 f(w m+1 ) b n f(w n ) = b m+1 f(w m+1 ) b n f(w n ) since the v i s are in the kernel. Thus, the f(w j ) s span the image. For linear independence, suppose Then 0 = b m+1 f(w m+1 ) b n f(w n ) 0 = f(b m+1 w m b n w n ) Then, b m+1 w m b n w n would be in the kernel of f, so would be a linear combination of the v i s, contradicting the fact that v 1,..., v m, w m+1,..., w n is a basis, unless all the b j s were 0. Thus, the f(w j ) are linearly independent, so are a basis for Imf. Exercises 5.[3.0.1] For subspaces V, W of a vector space over a field k, show that dim k V + dim k W = dim k (V + W ) + dim k (V W ) 5.[3.0.2] Given two bases e 1,..., e n and f 1,..., f n for a vector space V over a field k, show that there is a unique k-linear map T : V V such that T (e i ) = f i. 5.[3.0.3] Given a basis e 1,..., e n of a k-vectorspace V, and given arbitrary vectors w 1,..., w n in a k- vectorspace W, show that there is a unique k-linear map T : V W such that T e i = w i for all indices i.

6 90 Linear algebra I: dimension 5.[3.0.4] The space Hom k (V, W ) of k-linear maps from one k-vectorspace V to another, W, is a k- vectorspace under the operation (αt )(v) = α (T (v)) for α k and T Hom k (V, W ). Show that dim k Hom k (V, W ) = dim k V dim k W 5.[3.0.5] A flag V 1... V l of subspaces of a k-vectorspace V is simply a collection of subspaces satisfying the indicated inclusions. The type of the flag is the list of dimensions of the subspaces V i. Let W be a k-vectorspace, with a flag W 1... W l of the same type as the flag in V. Show that there exists a k-linear map T : V W such that T restricted to V i is an isomorphism V i W i. 5.[3.0.6] Let V 1 V l be a flag of subspace inside a finite-dimensional k-vectorspace V, and W 1... W l a flag inside another finite-dimensional k-vectorspace W. We do not suppose that the two flags are of the same type. Compute the dimension of the space of k-linear homomorphisms T : V W such that T V i W i.

### Mathematics 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

### THE 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

### I. 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

### Linear 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)),

### some algebra prelim solutions

some algebra prelim solutions David Morawski August 19, 2012 Problem (Spring 2008, #5). Show that f(x) = x p x + a is irreducible over F p whenever a F p is not zero. Proof. First, note that f(x) has no

### (January 14, 2009) End k (V ) End k (V/W )

(January 14, 29) [16.1] Let p be the smallest prime dividing the order of a finite group G. Show that a subgroup H of G of index p is necessarily normal. Let G act on cosets gh of H by left multiplication.

### 4.6 Null Space, Column Space, Row Space

NULL SPACE, COLUMN SPACE, ROW SPACE Null Space, Column Space, Row Space In applications of linear algebra, subspaces of R n typically arise in one of two situations: ) as the set of solutions of a linear

### MA106 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

### Linear 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

### Lecture Note on Linear Algebra 15. Dimension and Rank

Lecture Note on Linear Algebra 15. Dimension and Rank Wei-Shi Zheng, wszheng@ieee.org, 211 November 1, 211 1 What Do You Learn from This Note We still observe the unit vectors we have introduced in Chapter

### MATH10212 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

### Matrix 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

### LEARNING 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

### T ( 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)

### Sec 4.1 Vector Spaces and Subspaces

Sec 4. Vector Spaces and Subspaces Motivation Let S be the set of all solutions to the differential equation y + y =. Let T be the set of all 2 3 matrices with real entries. These two sets share many common

### An Advanced Course in Linear Algebra. Jim L. Brown

An Advanced Course in Linear Algebra Jim L. Brown July 20, 2015 Contents 1 Introduction 3 2 Vector spaces 4 2.1 Getting started............................ 4 2.2 Bases and dimension.........................

### ZORN S LEMMA AND SOME APPLICATIONS

ZORN S LEMMA AND SOME APPLICATIONS KEITH CONRAD 1. Introduction Zorn s lemma is a result in set theory that appears in proofs of some non-constructive existence theorems throughout mathematics. We will

### Metric Spaces. Chapter 7. 7.1. Metrics

Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some

### 3. 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,

### FUNCTIONAL 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

### GROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G.

Definition 1: GROUPS 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 the identity, such that (i) the

### Math 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

### 6. Fields I. 1. Adjoining things

6. Fields I 6.1 Adjoining things 6.2 Fields of fractions, fields of rational functions 6.3 Characteristics, finite fields 6.4 Algebraic field extensions 6.5 Algebraic closures 1. Adjoining things The general

### Definition 12 An alternating bilinear form on a vector space V is a map B : V V F such that

4 Exterior algebra 4.1 Lines and 2-vectors The time has come now to develop some new linear algebra in order to handle the space of lines in a projective space P (V ). In the projective plane we have seen

### MATH 110 Spring 2015 Homework 6 Solutions

MATH 110 Spring 2015 Homework 6 Solutions Section 2.6 2.6.4 Let α denote the standard basis for V = R 3. Let α = {e 1, e 2, e 3 } denote the dual basis of α for V. We would first like to show that β =

### Systems 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

### 1 Homework 1. [p 0 q i+j +... + p i 1 q j+1 ] + [p i q j ] + [p i+1 q j 1 +... + p i+j q 0 ]

1 Homework 1 (1) Prove the ideal (3,x) is a maximal ideal in Z[x]. SOLUTION: Suppose we expand this ideal by including another generator polynomial, P / (3, x). Write P = n + x Q with n an integer not

### CHAPTER 5: MODULAR ARITHMETIC

CHAPTER 5: MODULAR ARITHMETIC LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN 1. Introduction In this chapter we will consider congruence modulo m, and explore the associated arithmetic called

### (a) Write each of p and q as a polynomial in x with coefficients in Z[y, z]. deg(p) = 7 deg(q) = 9

Homework #01, due 1/20/10 = 9.1.2, 9.1.4, 9.1.6, 9.1.8, 9.2.3 Additional problems for study: 9.1.1, 9.1.3, 9.1.5, 9.1.13, 9.2.1, 9.2.2, 9.2.4, 9.2.5, 9.2.6, 9.3.2, 9.3.3 9.1.1 (This problem was not assigned

### MAT2400 Analysis I. A brief introduction to proofs, sets, and functions

MAT2400 Analysis I A brief introduction to proofs, sets, and functions In Analysis I there is a lot of manipulations with sets and functions. It is probably also the first course where you have to take

### SOLUTIONS TO PROBLEM SET 3

SOLUTIONS TO PROBLEM SET 3 MATTI ÅSTRAND The General Cubic Extension Denote L = k(α 1, α 2, α 3 ), F = k(a 1, a 2, a 3 ) and K = F (α 1 ). The polynomial f(x) = x 3 a 1 x 2 + a 2 x a 3 = (x α 1 )(x α 2

### Problems for Advanced Linear Algebra Fall 2012

Problems for Advanced Linear Algebra Fall 2012 Class will be structured around students presenting complete solutions to the problems in this handout. Please only agree to come to the board when you are

### Revision of ring theory

CHAPTER 1 Revision of ring theory 1.1. Basic definitions and examples In this chapter we will revise and extend some of the results on rings that you have studied on previous courses. A ring is an algebraic

### Chapter 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

### Math 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

### 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

### DIVISORS AND LINE BUNDLES

DIVISORS AND LINE BUNDLES TONY PERKINS 1. Cartier divisors An analytic hypersurface of M is a subset V M such that for each point x V there exists an open set U x M containing x and a holomorphic function

### S(A) X α for all α Λ. Consequently, S(A) X, by the definition of intersection. Therefore, X is inductive.

MA 274: Exam 2 Study Guide (1) Know the precise definitions of the terms requested for your journal. (2) Review proofs by induction. (3) Be able to prove that something is or isn t an equivalence relation.

### POWER SETS AND RELATIONS

POWER SETS AND RELATIONS L. MARIZZA A. BAILEY 1. The Power Set Now that we have defined sets as best we can, we can consider a sets of sets. If we were to assume nothing, except the existence of the empty

### MODULAR ARITHMETIC. a smallest member. It is equivalent to the Principle of Mathematical Induction.

MODULAR ARITHMETIC 1 Working With Integers The usual arithmetic operations of addition, subtraction and multiplication can be performed on integers, and the result is always another integer Division, on

### Vector Spaces II: Finite Dimensional Linear Algebra 1

John Nachbar September 2, 2014 Vector Spaces II: Finite Dimensional Linear Algebra 1 1 Definitions and Basic Theorems. For basic properties and notation for R N, see the notes Vector Spaces I. Definition

### Vector 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

### 4.5 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

### Linear 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

### Basic Terminology for Systems of Equations in a Nutshell. E. L. Lady. 3x 1 7x 2 +4x 3 =0 5x 1 +8x 2 12x 3 =0.

Basic Terminology for Systems of Equations in a Nutshell E L Lady A system of linear equations is something like the following: x 7x +4x =0 5x +8x x = Note that the number of equations is not required

### Sequences and Convergence in Metric Spaces

Sequences and Convergence in Metric Spaces Definition: A sequence in a set X (a sequence of elements of X) is a function s : N X. We usually denote s(n) by s n, called the n-th term of s, and write {s

### PROJECTIVE 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

### Introduction to finite fields

Introduction to finite fields Topics in Finite Fields (Fall 2013) Rutgers University Swastik Kopparty Last modified: Monday 16 th September, 2013 Welcome to the course on finite fields! This is aimed at

### 2.1 Functions. 2.1 J.A.Beachy 1. from A Study Guide for Beginner s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair

2.1 J.A.Beachy 1 2.1 Functions from A Study Guide for Beginner s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair 21. The Vertical Line Test from calculus says that a curve in the xy-plane

### Appendix A. Appendix. A.1 Algebra. Fields and Rings

Appendix A Appendix A.1 Algebra Algebra is the foundation of algebraic geometry; here we collect some of the basic algebra on which we rely. We develop some algebraic background that is needed in the text.

### Linearly 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

### We call this set an n-dimensional parallelogram (with one vertex 0). We also refer to the vectors x 1,..., x n as the edges of P.

Volumes of parallelograms 1 Chapter 8 Volumes of parallelograms In the present short chapter we are going to discuss the elementary geometrical objects which we call parallelograms. These are going to

### α = 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

### 6.2 Permutations continued

6.2 Permutations continued Theorem A permutation on a finite set A is either a cycle or can be expressed as a product (composition of disjoint cycles. Proof is by (strong induction on the number, r, of

### Math 225A, Differential Topology: Homework 3

Math 225A, Differential Topology: Homework 3 Ian Coley October 17, 2013 Problem 1.4.7. Suppose that y is a regular value of f : X Y, where X is compact and dim X = dim Y. Show that f 1 (y) is a finite

### Polynomial 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,

### Linear Dependence Tests

Linear Dependence Tests The book omits a few key tests for checking the linear dependence of vectors. These short notes discuss these tests, as well as the reasoning behind them. Our first test checks

### Riesz-Fredhölm Theory

Riesz-Fredhölm Theory T. Muthukumar tmk@iitk.ac.in Contents 1 Introduction 1 2 Integral Operators 1 3 Compact Operators 7 4 Fredhölm Alternative 14 Appendices 18 A Ascoli-Arzelá Result 18 B Normed Spaces

### Course 421: Algebraic Topology Section 1: Topological Spaces

Course 421: Algebraic Topology Section 1: Topological Spaces David R. Wilkins Copyright c David R. Wilkins 1988 2008 Contents 1 Topological Spaces 1 1.1 Continuity and Topological Spaces...............

### LINEAR 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,

### Finite dimensional C -algebras

Finite dimensional C -algebras S. Sundar September 14, 2012 Throughout H, K stand for finite dimensional Hilbert spaces. 1 Spectral theorem for self-adjoint opertors Let A B(H) and let {ξ 1, ξ 2,, ξ n

### ( ) 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

### Finite 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

### Course 221: Analysis Academic year , First Semester

Course 221: Analysis Academic year 2007-08, First Semester David R. Wilkins Copyright c David R. Wilkins 1989 2007 Contents 1 Basic Theorems of Real Analysis 1 1.1 The Least Upper Bound Principle................

### SOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties

SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 3 Fall 2008 III. Spaces with special properties III.1 : Compact spaces I Problems from Munkres, 26, pp. 170 172 3. Show that a finite union of compact subspaces

### Matrix Calculations: Kernels & Images, Matrix Multiplication

Matrix Calculations: Kernels & Images, Matrix Multiplication A. Kissinger (and H. Geuvers) Institute for Computing and Information Sciences Intelligent Systems Version: spring 2016 A. Kissinger Version:

### Weak topologies. David Lecomte. May 23, 2006

Weak topologies David Lecomte May 3, 006 1 Preliminaries from general topology In this section, we are given a set X, a collection of topological spaces (Y i ) i I and a collection of maps (f i ) i I such

### Linear 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.

### MATH 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

### Solutions 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

### (Basic definitions and properties; Separation theorems; Characterizations) 1.1 Definition, examples, inner description, algebraic properties

Lecture 1 Convex Sets (Basic definitions and properties; Separation theorems; Characterizations) 1.1 Definition, examples, inner description, algebraic properties 1.1.1 A convex set In the school geometry

### University 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

### 1 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

### Metric Spaces Joseph Muscat 2003 (Last revised May 2009)

1 Distance J Muscat 1 Metric Spaces Joseph Muscat 2003 (Last revised May 2009) (A revised and expanded version of these notes are now published by Springer.) 1 Distance A metric space can be thought of

### 8.1 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.

### Section 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

### Methods 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,

### Au = = = 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

### Lecture Notes 2: Matrices as Systems of Linear Equations

2: Matrices as Systems of Linear Equations 33A Linear Algebra, Puck Rombach Last updated: April 13, 2016 Systems of Linear Equations Systems of linear equations can represent many things You have probably

### (b) Prove that there is NO ordered basis γ of V s.t. Solution: (a) We have

Advanced Linear Algebra, Fall 2011. Solutions to midterm #1. 1. Let V = P 2 (R), the vector space of polynomials of degree 2 over R. Let T : V V be the differentiation map, that is, T (f(x)) = f (x). (a)

### by 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

### Elementary Number Theory We begin with a bit of elementary number theory, which is concerned

CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,

### GROUP 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

### MATH32062 Notes. 1 Affine algebraic varieties. 1.1 Definition of affine algebraic varieties

MATH32062 Notes 1 Affine algebraic varieties 1.1 Definition of affine algebraic varieties We want to define an algebraic variety as the solution set of a collection of polynomial equations, or equivalently,

### Structure of Measurable Sets

Structure of Measurable Sets In these notes we discuss the structure of Lebesgue measurable subsets of R from several different points of view. Along the way, we will see several alternative characterizations

### 7 - Linear Transformations

7 - Linear Transformations Mathematics has as its objects of study sets with various structures. These sets include sets of numbers (such as the integers, rationals, reals, and complexes) whose structure

### Orthogonal 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

### Chapter 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

### 4: 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:

### POLYTOPES WITH MASS LINEAR FUNCTIONS, PART I

POLYTOPES WITH MASS LINEAR FUNCTIONS, PART I DUSA MCDUFF AND SUSAN TOLMAN Abstract. We analyze mass linear functions on simple polytopes, where a mass linear function is an affine function on whose value

### Group Theory. Contents

Group Theory Contents Chapter 1: Review... 2 Chapter 2: Permutation Groups and Group Actions... 3 Orbits and Transitivity... 6 Specific Actions The Right regular and coset actions... 8 The Conjugation

### Classical Analysis I

Classical Analysis I 1 Sets, relations, functions A set is considered to be a collection of objects. The objects of a set A are called elements of A. If x is an element of a set A, we write x A, and if

### Sets and functions. {x R : x > 0}.

Sets and functions 1 Sets The language of sets and functions pervades mathematics, and most of the important operations in mathematics turn out to be functions or to be expressible in terms of functions.

### CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE

CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE With the notion of bijection at hand, it is easy to formalize the idea that two finite sets have the same number of elements: we just need to verify

### These 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

### GROUPS ACTING ON A SET

GROUPS ACTING ON A SET MATH 435 SPRING 2012 NOTES FROM FEBRUARY 27TH, 2012 1. Left group actions Definition 1.1. Suppose that G is a group and S is a set. A left (group) action of G on S is a rule for

### Geometric Transformations

Geometric Transformations Definitions Def: f is a mapping (function) of a set A into a set B if for every element a of A there exists a unique element b of B that is paired with a; this pairing is denoted

### 1 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