DOCSLIB.ORG

Chapter 4 Isomorphism and Coordinates

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Chapter 4 Isomorphism and Coordinates Chapter 4 Isomorphism and Coordinates Recall that a vector space isomorphism is a linear map that is both one-to-one and onto. Such a map preserves every aspect of the vector space structure. In other words, if L: V → W is an isomorphism, then any true statement you can say about V using abstract vector notation, vector addition, and scalar multiplication, willtransfertoatruestatement about W when L is applied to the entire statement. We make this more precise with some examples. Example. If L: V → W is an isomorphism, then the set {v1,...,vn} is linearly indepen- dent in V if and only if the set {L(v1),...,L(vn)} is linearly independent in W.The dimension of the subspace spanned by the first set equals the dimension of the subset spanned by the second set. In particular, the dimension of V equals that of W. This last statement about dimension is only one part of a more fundamental fact. Theorem 4.0.1. Suppose V is a finite-dimensional vector space. Then V is isomorphic to W if and only if dim V = dim W. Proof. Suppose that V and W are isomorphic, and let L: V → W be an isomorphism. Then L is one-to-one, so dim ker L = 0. Since L is onto, we also have dim imL = dim W. Plugging these into the rank-nullity theorem for L shows then that dim V = dim W. Now suppose that dim V = dim W = n,andchoosebases{v1,...,vn} and {w1,...,wn} for V and W,respectively.Foranyvectorv in V,wewritev = a1v1 + ···+ anvn,and define L(v)=L(a1v1 + ···+ anvn)=a1w1 + ···+ anwn. We claim that L is linear, one-to-one, and onto. (Proof omitted.) In particular, and 2–dimensional real vector space is necessarily isomorphic to R2,for example. This helps to explain why so many problems in these other spaces ended up reducing to solving systems of equations just like those we saw in Rn. Looking at the proof, we see that isomorphisms are constructed by sending bases to bases. In particular, there is a different isomorphism V → W for each choice of basis for V and for W. 27 28 CHAPTER 4. ISOMORPHISM AND COORDINATES One special case of this is when we look at isomorphisms V → V.Suchanisomor- phism is called a change of coordinates. If S = {v1,...,vn} is a basis for V,wesaythen-tuple (a1,...,an) is the coordinate vector of v with respect to S if v = a1v + ···+ anvn.Wedenotethisvectoras[v]S. Example. Find the coordinates for (1, 3) with respect to the basis S = {(1, 1), (−1, 1)}. We set (1, 3)=a(1, 1)+b(−1, 1),whichleadstotheequationsa − b = 1 and a + b = 3. This system has solution a = 2, b = 1. Thus (1, 3)=2(1, 1)+1(−1, 1), so that [(1, 3)]S = (2, 1). Example. Find the coordinates for t2 + 3t + 2withrespecttothebasisS = {t2 + 1, t + 1, t − 1}.Wesett2 + 3t + 2 = a(t2 + 1)+b(t + 1)+c(t − 1).Collectingliketermsgives t2 + 3t + 2 = at2 +(b + c)t +(a + b − c).Thisleadstothesystemofequations a = 1 b + c = 3 a + b − c = 2 The solution is a = 1, b = 2, c = 1. Thus we have t2 + 3t + 2 = 1(t2 + 1)+2(t + 1)+ 2 1(t − 1),sothat[t + 3t + 2]S =(1, 2, 1). Note that for any vector v in an n–dimensional vector space V and for any basis S for n V, the coordinate vector [v]S is an element of R . Proposition 4.0.2. For any basis S for an n–dimensional vector space V,thecorrespondence n v #→ [v]S is an isomorphism from V to R . Corollary 4.0.3. Every n–dimensional vector space over a R is isomorphic to Rn. Chapter 5 Linear Maps Rn → Rm Since every finite-dimensional vector space over R is isomorphic to Rn, any problem we have in such a vector space that can be expressed entirely in terms of vector operations can be tranferred to one in Rn.Sinceourultimategoalistounderstandlinearmaps V → W,wewillfocusoureffortsonunderstandinglinearmapsRn → Rm,without worrying about expressing things in abstract terms. Remark. Unlike any previous section, we focus specifically on Rn in this chapter. To emphasize the distinction, we use x to denote an arbitrary vector in Rn. 5.1 Linear maps from Rn to R We’ve already seen above that the linear maps R → R are precisely those of the form L(x)=ax for some real number a.Forthenextstep,weallowourdomaintohave multiple dimensions, but insist that our target space be R.Wewilldiscoverthatlinear maps L: Rn → R are already familiar to us. Theorem 5.1.1. If L: Rn → R is a linear map, then there is some vector m such that L(x)=a · x. n Proof. For j = 1, . , n,wesetej equal to the jth standard basis vector in R .Seta = (a1,...,an),whereeachaj = L(ej),andconsideranarbitraryvectorx =(x1,...,xn) in Rn.Wecompute L(x)=L(x1e1 + ···+ xnen)=x1L(e1)+···+ xnL(en)=x1a1 + ···+ xnan = x · a. Remark. Wait, didn’t we say that we weren’t going to think about dot products? Then we would be studying inner product spaces rather than vector spaces! Yes, and that’s still true. Within a given vector space, we will not be performing any dot products, and so in particular will never speak of length or angle. And infactourdefinitionof linear map did not use the notion of dot product; it used only vector addition and scalar multiplication. What we’ve shown is that every linear map from Rn to R has the form f (x1, x2,...,xn)=a1x1 + ···+ anxn 29 30 CHAPTER 5. LINEAR MAPS RN → RM for some fixed real numbers a1,...,an.Itjustsohappensthatwehaveanameforthis type of operation, and we call it the dot product, but this is just a convenient way to explain what linear maps do; we’re not studying the algebraic or geometric properties of the dot product in Rn. 5.2 Linear Maps Rn → Rm One of the first things you learn in vector calculus is that functions with multiple outputs can be thought of as a list of functions with one output. Thus given an arbitrary func- 2 3 tion f : R → R ,say,wethinkofitasf (x, y)=(f1(x, y), f2(x, y), f3(x, y)),whereeach 2 1 component function fj is a map R → R .Wethusexpecttofindthatlinearmapsfrom Rn to Rm are those whose component functions are linear maps from Rn to R, which we saw in the last section are just dot products. This is the content of the following. Theorem 5.2.1. The function L: Rn → Rm is linear if and only if each component function n Lj : R → R is linear. Proof. Omitted. Thus any linear map Rn → Rm is built up from a bunch of dot products in each component. In the next section we will make use of this fact to come up with a nice way to present linear maps. 5.3 Matrices There are many ways to write vectors in Rn.Forexample,thesamevectorinR3 can be represented as 3 3i + 2j − 4k, $3, 2, −4%, (3, 2, −4), [3, 2, −4], 2 . −4 We will focus on these last two for the time being. In particular, whenever we have a dot product x · y of two vectors x and y (in that order), we will write the first as a row in square brackets and the second as a column in square brackets. Thus we have, for example, 2 123 3 = 2 + 6 − 12 = −4. −4 ) * Note that we are also avoiding commas in the row vector. 5.3. MATRICES 31 Now suppose L is an arbitrary linear map from Rn to R.Thengiveninputvectorx, L(x) is the dot product a · x for some fixed vector a.Thuswemaywrite x1 x1 x x 2 2 L . = a1 a2 ··· an . x ) * x n n Now suppose L is a linear map from Rn to Rm,andtheith component functions is the dot product with ai.Thewecanwrite x1 a11 a12 ··· a1n x1 a1 · x x2 a21 a22 ··· a2n x2 a2 · x L . = . . = . . ··· . x a a ··· a x a · x n m1 m2 mn n m Thus we can think of any linear map from Rn to Rm as multiplication by a matrix, assuming we define multiplication in exactly this way. Definition 5.3.1. If A =(aij) is an m × n matrix and x is an n × 1columnvector,the product Ax is defined to be the m × 1 column vector whose ith entry is the dot product of the ith row of A with x. Thus we are led to the fortuitous observation that every linear map L: Rn → Rm has the form L(x)=Ax for some m × n matrix A.ThuslinearmapsfromR to itself are just multiplication by a 1 × 1matrix;i.e.,multiplicationbyaconstant.Thisagreeswith what we saw earlier. We now note an important fact about compositions of linear maps. Theorem 5.3.2. Suppose L: Rn → Rm and T : Rm → Rp are linear maps. Then the composition T ◦ L: Rn → Rp is a linear map. Suppose L is represented by the m × n matrix A and T is represented by the p × m matrix B.BecauseT ◦ L is also linear, it is represented by some p × n matrix C.Wenow show how to construct C from A and B. We begin with a motivating example. Suppose L maps from R2 to R2,asdoesT,and suppose L dots with a =(a1, a2, ) and b =(b1, b2) while T dots with c =(c1, c2) and d =(d1, d2).Then a · x a x + a x c (a x + a x )+c (b x + b x ) T ◦ L(x)=T = T 1 1 2 2 = 1 1 1 2 2 2 1 1 2 2 b · x b x + b x d (a x + a x )+d (b x + b x ) 34 56 34 1 1 2 256 4 1 1 1 2 2 2 1 1 2 2 5 c a + c b c a + c b x = 1 1 2 1 1 2 2 2 1 d a + d b d a + d b x 4 1 1 2 1 1 2 2 254 25 32 CHAPTER 5.
Load more

Recommended publications
  • Chapter 4. Homomorphisms and Isomorphisms of Groups
    Chapter 4. Homomorphisms and Isomorphisms of Groups 4.1 Note: We recall the following terminology. Let X and Y be sets. When we say that f is a function or a map from X to Y , written f : X ! Y , we mean that for every x 2 X there exists a unique corresponding element y = f(x) 2 Y . The set X is called the domain of f and the range or image of f is the set Image(f) = f(X) = f(x) x 2 X . For a set A ⊆ X, the image of A under f is the set f(A) = f(a) a 2 A and for a set −1 B ⊆ Y , the inverse image of B under f is the set f (B) = x 2 X f(x) 2 B . For a function f : X ! Y , we say f is one-to-one (written 1 : 1) or injective when for every y 2 Y there exists at most one x 2 X such that y = f(x), we say f is onto or surjective when for every y 2 Y there exists at least one x 2 X such that y = f(x), and we say f is invertible or bijective when f is 1:1 and onto, that is for every y 2 Y there exists a unique x 2 X such that y = f(x). When f is invertible, the inverse of f is the function f −1 : Y ! X defined by f −1(y) = x () y = f(x). For f : X ! Y and g : Y ! Z, the composite g ◦ f : X ! Z is given by (g ◦ f)(x) = g(f(x)).
    [Show full text]
  • The General Linear Group
    18.704 Gabe Cunningham 2/18/05 [email protected] The General Linear Group Definition: Let F be a field. Then the general linear group GLn(F ) is the group of invert- ible n × n matrices with entries in F under matrix multiplication. It is easy to see that GLn(F ) is, in fact, a group: matrix multiplication is associative; the identity element is In, the n × n matrix with 1’s along the main diagonal and 0’s everywhere else; and the matrices are invertible by choice. It’s not immediately clear whether GLn(F ) has infinitely many elements when F does. However, such is the case. Let a ∈ F , a 6= 0. −1 Then a · In is an invertible n × n matrix with inverse a · In. In fact, the set of all such × matrices forms a subgroup of GLn(F ) that is isomorphic to F = F \{0}. It is clear that if F is a finite field, then GLn(F ) has only finitely many elements. An interesting question to ask is how many elements it has. Before addressing that question fully, let’s look at some examples. ∼ × Example 1: Let n = 1. Then GLn(Fq) = Fq , which has q − 1 elements. a b Example 2: Let n = 2; let M = ( c d ). Then for M to be invertible, it is necessary and sufficient that ad 6= bc. If a, b, c, and d are all nonzero, then we can fix a, b, and c arbitrarily, and d can be anything but a−1bc. This gives us (q − 1)3(q − 2) matrices.
    [Show full text]
  • MA 242 LINEAR ALGEBRA C1, Solutions to Second Midterm Exam
    MA 242 LINEAR ALGEBRA C1, Solutions to Second Midterm Exam Prof. Nikola Popovic, November 9, 2006, 09:30am - 10:50am Problem 1 (15 points). Let the matrix A be given by 1 −2 −1 2 −1 5 6 3 : 5 −4 5 4 5 (a) Find the inverse A−1 of A, if it exists. (b) Based on your answer in (a), determine whether the columns of A span R3. (Justify your answer!) Solution. (a) To check whether A is invertible, we row reduce the augmented matrix [A I3]: 1 −2 −1 1 0 0 1 −2 −1 1 0 0 2 −1 5 6 0 1 0 3 ∼ : : : ∼ 2 0 3 5 1 1 0 3 : 5 −4 5 0 0 1 0 0 0 −7 −2 1 4 5 4 5 Since the last row in the echelon form of A contains only zeros, A is not row equivalent to I3. Hence, A is not invertible, and A−1 does not exist. (b) Since A is not invertible by (a), the Invertible Matrix Theorem says that the columns of A cannot span R3. Problem 2 (15 points). Let the vectors b1; : : : ; b4 be defined by 3 2 −1 0 0 5 1 0 −1 1 0 1 1 0 0 1 b1 = ; b2 = ; b3 = ; and b4 = : −2 −5 3 0 B C B C B C B C B 4 C B 7 C B 0 C B −3 C @ A @ A @ A @ A (a) Determine if the set B = fb1; b2; b3; b4g is linearly independent by computing the determi- nant of the matrix B = [b1 b2 b3 b4].
    [Show full text]
  • Categories, Functors, and Natural Transformations I∗
    Lecture 2: Categories, functors, and natural transformations I∗ Nilay Kumar June 4, 2014 (Meta)categories We begin, for the moment, with rather loose definitions, free from the technicalities of set theory. Definition 1. A metagraph consists of objects a; b; c; : : :, arrows f; g; h; : : :, and two operations, as follows. The first is the domain, which assigns to each arrow f an object a = dom f, and the second is the codomain, which assigns to each arrow f an object b = cod f. This is visually indicated by f : a ! b. Definition 2. A metacategory is a metagraph with two additional operations. The first is the identity, which assigns to each object a an arrow Ida = 1a : a ! a. The second is the composition, which assigns to each pair g; f of arrows with dom g = cod f an arrow g ◦ f called their composition, with g ◦ f : dom f ! cod g. This operation may be pictured as b f g a c g◦f We require further that: composition is associative, k ◦ (g ◦ f) = (k ◦ g) ◦ f; (whenever this composition makese sense) or diagrammatically that the diagram k◦(g◦f)=(k◦g)◦f a d k◦g f k g◦f b g c commutes, and that for all arrows f : a ! b and g : b ! c, we have 1b ◦ f = f and g ◦ 1b = g; or diagrammatically that the diagram f a b f g 1b g b c commutes. ∗This talk follows [1] I.1-4 very closely. 1 Recall that a diagram is commutative when, for each pair of vertices c and c0, any two paths formed from direct edges leading from c to c0 yield, by composition of labels, equal arrows from c to c0.
    [Show full text]
  • Irreducible Representations of Finite Monoids
    U.U.D.M. Project Report 2019:11 Irreducible representations of finite monoids Christoffer Hindlycke Examensarbete i matematik, 30 hp Handledare: Volodymyr Mazorchuk Examinator: Denis Gaidashev Mars 2019 Department of Mathematics Uppsala University Irreducible representations of finite monoids Christoffer Hindlycke Contents Introduction 2 Theory 3 Finite monoids and their structure . .3 Introductory notions . .3 Cyclic semigroups . .6 Green’s relations . .7 von Neumann regularity . 10 The theory of an idempotent . 11 The five functors Inde, Coinde, Rese,Te and Ne ..................... 11 Idempotents and simple modules . 14 Irreducible representations of a finite monoid . 17 Monoid algebras . 17 Clifford-Munn-Ponizovski˘ıtheory . 20 Application 24 The symmetric inverse monoid . 24 Calculating the irreducible representations of I3 ........................ 25 Appendix: Prerequisite theory 37 Basic definitions . 37 Finite dimensional algebras . 41 Semisimple modules and algebras . 41 Indecomposable modules . 42 An introduction to idempotents . 42 1 Irreducible representations of finite monoids Christoffer Hindlycke Introduction This paper is a literature study of the 2016 book Representation Theory of Finite Monoids by Benjamin Steinberg [3]. As this book contains too much interesting material for a simple master thesis, we have narrowed our attention to chapters 1, 4 and 5. This thesis is divided into three main parts: Theory, Application and Appendix. Within the Theory chapter, we (as the name might suggest) develop the necessary theory to assist with finding irreducible representations of finite monoids. Finite monoids and their structure gives elementary definitions as regards to finite monoids, and expands on the basic theory of their structure. This part corresponds to chapter 1 in [3]. The theory of an idempotent develops just enough theory regarding idempotents to enable us to state a key result, from which the principal result later follows almost immediately.
    [Show full text]
  • Limits Commutative Algebra May 11 2020 1. Direct Limits Definition 1
    Limits Commutative Algebra May 11 2020 1. Direct Limits Definition 1: A directed set I is a set with a partial order ≤ such that for every i; j 2 I there is k 2 I such that i ≤ k and j ≤ k. Let R be a ring. A directed system of R-modules indexed by I is a collection of R modules fMi j i 2 Ig with a R module homomorphisms µi;j : Mi ! Mj for each pair i; j 2 I where i ≤ j, such that (i) for any i 2 I, µi;i = IdMi and (ii) for any i ≤ j ≤ k in I, µi;j ◦ µj;k = µi;k. We shall denote a directed system by a tuple (Mi; µi;j). The direct limit of a directed system is defined using a universal property. It exists and is unique up to a unique isomorphism. Theorem 2 (Direct limits). Let fMi j i 2 Ig be a directed system of R modules then there exists an R module M with the following properties: (i) There are R module homomorphisms µi : Mi ! M for each i 2 I, satisfying µi = µj ◦ µi;j whenever i < j. (ii) If there is an R module N such that there are R module homomorphisms νi : Mi ! N for each i and νi = νj ◦µi;j whenever i < j; then there exists a unique R module homomorphism ν : M ! N, such that νi = ν ◦ µi. The module M is unique in the sense that if there is any other R module M 0 satisfying properties (i) and (ii) then there is a unique R module isomorphism µ0 : M ! M 0.
    [Show full text]
  • Homomorphisms and Isomorphisms
    Lecture 4.1: Homomorphisms and isomorphisms Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4120, Modern Algebra M. Macauley (Clemson) Lecture 4.1: Homomorphisms and isomorphisms Math 4120, Modern Algebra 1 / 13 Motivation Throughout the course, we've said things like: \This group has the same structure as that group." \This group is isomorphic to that group." However, we've never really spelled out the details about what this means. We will study a special type of function between groups, called a homomorphism. An isomorphism is a special type of homomorphism. The Greek roots \homo" and \morph" together mean \same shape." There are two situations where homomorphisms arise: when one group is a subgroup of another; when one group is a quotient of another. The corresponding homomorphisms are called embeddings and quotient maps. Also in this chapter, we will completely classify all finite abelian groups, and get a taste of a few more advanced topics, such as the the four \isomorphism theorems," commutators subgroups, and automorphisms. M. Macauley (Clemson) Lecture 4.1: Homomorphisms and isomorphisms Math 4120, Modern Algebra 2 / 13 A motivating example Consider the statement: Z3 < D3. Here is a visual: 0 e 0 7! e f 1 7! r 2 2 1 2 7! r r2f rf r2 r The group D3 contains a size-3 cyclic subgroup hri, which is identical to Z3 in structure only. None of the elements of Z3 (namely 0, 1, 2) are actually in D3. When we say Z3 < D3, we really mean is that the structure of Z3 shows up in D3.
    [Show full text]
  • Friday September 20 Lecture Notes
    Friday September 20 Lecture Notes 1 Functors Definition Let C and D be categories. A functor (or covariant) F is a function that assigns each C 2 Obj(C) an object F (C) 2 Obj(D) and to each f : A ! B in C, a morphism F (f): F (A) ! F (B) in D, satisfying: For all A 2 Obj(C), F (1A) = 1FA. Whenever fg is defined, F (fg) = F (f)F (g). e.g. If C is a category, then there exists an identity functor 1C s.t. 1C(C) = C for C 2 Obj(C) and for every morphism f of C, 1C(f) = f. For any category from universal algebra we have \forgetful" functors. e.g. Take F : Grp ! Cat of monoids (·; 1). Then F (G) is a group viewed as a monoid and F (f) is a group homomorphism f viewed as a monoid homomor- phism. e.g. If C is any universal algebra category, then F : C! Sets F (C) is the underlying sets of C F (f) is a morphism e.g. Let C be a category. Take A 2 Obj(C). Then if we define a covariant Hom functor, Hom(A; ): C! Sets, defined by Hom(A; )(B) = Hom(A; B) for all B 2 Obj(C) and f : B ! C, then Hom(A; )(f) : Hom(A; B) ! Hom(A; C) with g 7! fg (we denote Hom(A; ) by f∗). Let us check if f∗ is a functor: Take B 2 Obj(C). Then Hom(A; )(1B) = (1B)∗ : Hom(A; B) ! Hom(A; B) and for g 2 Hom(A; B), (1B)∗(g) = 1Bg = g.
    [Show full text]
  • 7.3 Isomorphisms and Composition
    392 Linear Transformations 7.3 Isomorphisms and Composition Often two vector spaces can consist of quite different types of vectors but, on closer examination, turn out to be the same underlying space displayed in different symbols. For example, consider the spaces 2 R = (a, b) a, b R and P1 = a + bx a, b R { | ∈ } { | ∈ } Compare the addition and scalar multiplication in these spaces: (a, b)+(a1, b1)=(a + a1, b + b1) (a + bx)+(a1 + b1x)=(a + a1)+(b + b1)x r(a, b)=(ra, rb) r(a + bx)=(ra)+(rb)x Clearly these are the same vector space expressed in different notation: if we change each (a, b) in R2 to 2 a + bx, then R becomes P1, complete with addition and scalar multiplication. This can be expressed by 2 noting that the map (a, b) a + bx is a linear transformation R P1 that is both one-to-one and onto. In this form, we can describe7→ the general situation. → Definition 7.4 Isomorphic Vector Spaces A linear transformation T : V W is called an isomorphism if it is both onto and one-to-one. The vector spaces V and W are said→ to be isomorphic if there exists an isomorphism T : V W, and → we write V ∼= W when this is the case. Example 7.3.1 The identity transformation 1V : V V is an isomorphism for any vector space V . → Example 7.3.2 T If T : Mmn Mnm is defined by T (A) = A for all A in Mmn, then T is an isomorphism (verify).
    [Show full text]
  • Monomorphism - Wikipedia, the Free Encyclopedia
    Monomorphism - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Monomorphism Monomorphism From Wikipedia, the free encyclopedia In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation . In the more general setting of category theory, a monomorphism (also called a monic morphism or a mono) is a left-cancellative morphism, that is, an arrow f : X → Y such that, for all morphisms g1, g2 : Z → X, Monomorphisms are a categorical generalization of injective functions (also called "one-to-one functions"); in some categories the notions coincide, but monomorphisms are more general, as in the examples below. The categorical dual of a monomorphism is an epimorphism, i.e. a monomorphism in a category C is an epimorphism in the dual category Cop. Every section is a monomorphism, and every retraction is an epimorphism. Contents 1 Relation to invertibility 2 Examples 3 Properties 4 Related concepts 5 Terminology 6 See also 7 References Relation to invertibility Left invertible morphisms are necessarily monic: if l is a left inverse for f (meaning l is a morphism and ), then f is monic, as A left invertible morphism is called a split mono. However, a monomorphism need not be left-invertible. For example, in the category Group of all groups and group morphisms among them, if H is a subgroup of G then the inclusion f : H → G is always a monomorphism; but f has a left inverse in the category if and only if H has a normal complement in G.
    [Show full text]
  • Group Properties and Group Isomorphism
    GROUP PROPERTIES AND GROUP ISOMORPHISM Evelyn. M. Manalo Mathematics Honors Thesis University of California, San Diego May 25, 2001 Faculty Mentor: Professor John Wavrik Department of Mathematics GROUP PROPERTIES AND GROUP ISOMORPHISM I n t r o d u c t i o n T H E I M P O R T A N C E O F G R O U P T H E O R Y is relevant to every branch of Mathematics where symmetry is studied. Every symmetrical object is associated with a group. It is in this association why groups arise in many different areas like in Quantum Mechanics, in Crystallography, in Biology, and even in Computer Science. There is no such easy definition of symmetry among mathematical objects without leading its way to the theory of groups. In this paper we present the first stages of constructing a systematic method for classifying groups of small orders. Classifying groups usually arise when trying to distinguish the number of non-isomorphic groups of order n. This paper arose from an attempt to find a formula or an algorithm for classifying groups given invariants that can be readily determined without any other known assumptions about the group. This formula is very useful if we want to know if two groups are isomorphic. Mathematical objects are considered to be essentially the same, from the point of view of their algebraic properties, when they are isomorphic. When two groups Γ and Γ’ have exactly the same group-theoretic structure then we say that Γ is isomorphic to Γ’ or vice versa.
    [Show full text]
  • Notes on Change of Bases Northwestern University, Summer 2014
    Notes on Change of Bases Northwestern University, Summer 2014 Let V be a finite-dimensional vector space over a field F, and let T be a linear operator on V . Given a basis (v1; : : : ; vn) of V , we've seen how we can define a matrix which encodes all the information about T as follows. For each i, we can write T vi = a1iv1 + ··· + anivn 2 for a unique choice of scalars a1i; : : : ; ani 2 F. In total, we then have n scalars aij which we put into an n × n matrix called the matrix of T relative to (v1; : : : ; vn): 0 1 a11 ··· a1n B . .. C M(T )v := @ . A 2 Mn;n(F): an1 ··· ann In the notation M(T )v, the v showing up in the subscript emphasizes that we're taking this matrix relative to the specific bases consisting of v's. Given any vector u 2 V , we can also write u = b1v1 + ··· + bnvn for a unique choice of scalars b1; : : : ; bn 2 F, and we define the coordinate vector of u relative to (v1; : : : ; vn) as 0 1 b1 B . C n M(u)v := @ . A 2 F : bn In particular, the columns of M(T )v are the coordinates vectors of the T vi. Then the point of the matrix M(T )v is that the coordinate vector of T u is given by M(T u)v = M(T )vM(u)v; so that from the matrix of T and the coordinate vectors of elements of V , we can in fact reconstruct T itself.
    [Show full text]