Chapter 1. Group Representations. Definition 1.1 A representation α of a group G in a vector space V over k is defined by a homomorphism

Save this PDF as:

Size: px
Start display at page:

Download "Chapter 1. Group Representations. Definition 1.1 A representation α of a group G in a vector space V over k is defined by a homomorphism"

Transcription

1 Chapter 1 Group Representations Definition 1.1 A representation α of a group G in a vector space V over k is defined by a homomorphism α : G GL(V ). The degree of the representation is the dimension of the vector space: deg α = dim k V. Remarks: 1. Recall that GL(V ) the general linear group on V is the group of invertible (or non-singular) linear maps t : V V. 2. We shall be concerned almost exclusively with representations of finite degree, that is, in finite-dimensional vector spaces; and these will almost always be vector spaces over R or C. Therefore, to avoid repetition, let us agree to use the term representation to mean representation of finite degree over R or C, unless the contrary is explicitly stated. Furthermore, in this first Part we shall be concerned almost exclusively with finite groups; so let us also agree that the term group will mean finite group in this Part, unless the contrary is stated. 3. Suppose {e 1,..., e n } is a basis for V. Then each linear map t : V V is defined (with respect to this basis) by an n n-matrix T. Explicitly, te j = i T ij e i ; 424 I 1 1

2 424 I 1 2 or in terms of coordinates, x 1. x n T x 1. x n Thus a representation in V can be defined by a homomorphism α : G GL(n, k), where GL(n, k) denotes the group of non-singular n n-matrices over k. In other words, α is defined by giving matrices A(g) for each g G, satisfying the conditions A(gh) = A(g)A(h) for all g, h G; and also A(e) = I. 4. There is another way of looking at group representations which is almost always more fruitful than the rather abstract definition we started with. Recall that a group is said to act on the set X if we have a map satisfying (a) (gh)x) = g(hx), (b) ex = x. G X X : (g, x) gx Now suppose X = V is a vector space. Then we can say that G acts linearly on V if in addition (c) g(u + v) = gu + gv, (d) g(ρv) = ρ(gv). Each representation α of G in V defines a linear action of G on V, by gv = α(g)v; and every such action arises from a representation in this way. Thus the notions of representation and linear action are completely equivalent. We can use whichever we find more convenient in a given case.

3 424 I There are two other ways of looking at group representations, completely equivalent to the definition but expressing slightly different points of view. Firstly, we may speak of the vector space V, with the action of G on it. as a G-space. For those familiar with category theory, this would be the categorical approach. Representation theory, from this point of view, is the study of the category of G-spaces and G-maps, where a G-map t : U V from one G-space to another is a linear map preserving the action of G, ie satisfying t(gu) = g(tu) (g G, u U). 6. Secondly, and finally, mathematical physicists often speak strikingly of the vector space V carrying the representation α. Examples: 1. Recall that the dihedral group D 4 is the symmetry group of a square ABCD y B A O x C D Figure 1.1: The natural representation of D 4 (Figure??). Let us take coordinates Ox, Oy as shown through the centre O of the square. Then D 4 = {e, r, r 2, r 3, c, d, x, y}, where r is the rotation about O through π/2 (sending A to B), while c, d, x, y are the reflections in AC, BD, Ox, Oy respectively.

4 424 I 1 4 By definition a symmetry g D 4 is an isometry of the plane E 2 sending the square into itself. Evidently g must send O into itself, and so gives rise to a linear map A(g) : R 2 R 2. The map g A(g) GL(2, R) defines a 2-dimensional representation ρ of D 4 over R. We may describe this as the natural 2-dimensional representation of D 4. (Evidently the symmetry group G of any bounded subset S E n will have a similar natural representation in R n.) The representation ρ is given in matrix terms by ( ) ( ) ( ) ( ) e, r, r , r 0 1 3, 1 0 ( ) ( ) ( ) ( ) c, d, x, y Each group relation is represented in a corresponding matrix equation, eg ( ) ( ) ( ) cd = r 2 = = The representation ρ is faithful, ie the homomorphism defining it is injective. Thus a relation holds in D 4 if and only if the corresponding matrix equation is true. However, representations are not necessarily faithful, and in general the implication is only one way. Every finite-dimensional representation can be expressed in matrix form in this way, after choosing a basis for the vector space carrying the representation. However, while such matrix representations are reassuringly concrete, they are impractical except in the lowest dimensions. Better just to keep at the back of one s mind that a representation could be expressed in this way. 2. Suppose G acts on the set X: (g, x) gx. Let denote the space of maps C(X) = C(X, k) f : X k.

5 424 I 1 5 Then G acts linearly on C(X) and so defines a representation ρ of G by gf(x) = f(g 1 x). (We need g 1 rather than g on the right to satisfy the rule It is fortunate that the relation g(hf) = (gh)f. (gh) 1 = h 1 g 1 enables us to correct the order reversal. We shall often have occasion to take advantage of this, particularly when dealing as here with spaces of functions.) Now suppose that X is finite; say Then X = {x 1,..., x n }. deg ρ = n = X, the number of elements in X. For the functions { 1 if x = y, e y (x) = 0 otherwise. (ie the characteristic functions of the 1-point subsets) form a basis for C(X). Also ge x = e gx, since ge y (x) = e y (g 1 x) 1 if g 1 x = y = 0 if g 1 x y 1 if x = gy = 0 if x gy Thus g P (g)

6 424 I 1 6 where P = P (g) is the matrix with entries { 1 if y = gx, P xy = 0 otherwise. Notice that P is a permutation matrix, ie there is just one 1 in each row and column, all other entries being 0. We call a representation that arises from the action of a group on a set in this way a permutational representation. As an illustration, consider the natural action of S(3) on the set {a, b, c}. This yields a 3-dimensional representation ρ of S(3), under which (abc) (ab) (These two instances actually define the representation, since (abc) and (ab) generate S(3).) 3. A 1-dimensional representation α of a group G over k = R or C is just a homomorphism α : G k, where k denotes the multiplicative group on the set k \ {0}. For GL(1, k) = k, since we can identify the 1 1-matrix [x] with its single entry x. We call the 1-dimensional representation defined by the identity homomorphism g 1 (for all g G) the trivial representation of G, and denote it by 1. In a 1-dimensional representation, each group element is represented by a number. Since these numbers commute, the study of 1-dimensional representations is much simpler than those of higher dimension. In general, when investigating the representations of a group G, we start by determining all its 1-dimensional representations. Recall that two elements g, h G are said to be conjugate if h = xgx 1

7 424 I 1 7 for some third element x G. Suppose α is a 1-dimensional representation of G. Then α(h) = α(x)α(g)α(x 1 ) = α(x)α(g)α(x) 1 = α(g)α(x)α(x) 1 = α(g), since the numbers α(x), α(g) commute. It follows that a 1-dimensional representation is constant on each conjugacy class of G. Consider the group S 3. This has 3 classes (we shall usually abbreviate conjugacy class to class): {1}, {(abc), (acb)}, {(bc), (ca), (ab)}. Let us write Then (assuming k = C) s = (abc), t = (bc). But It follows that s 3 = 1 = α(s) 3 = 1 = α(s) = 1, ω or ω 2, t 2 = 1 = α(s) 2 = 1 = α(t) = ±1. from which we deduce that tst 1 = s 2. α(t)α(s)α(t) 1 = α(s) 2, α(s) = 1. It follows that S 3 has just two 1-dimensional representations: the trivial representation 1 : g 1, and the parity representation { 1 if g is even, ɛ : g 1 if g is odd. 4. The corresponding result is true for all the symmetric groups S n (for n 2); S n has just two 1-dimensional representations, the trivial representation 1 and the parity representation ɛ. To see this, let us recall two facts about S n.

8 424 I 1 8 (a) The transpositions τ = (xy) generate S n, ie each permutation g S n is expressible (not uniquely) as a product of transpositions g = τ 1 τ r. (b) The transpositions are all conjugate. (This is a particular case of the general fact that two permutations in S n are conjugate if and only if they are of the same cyclic type, ie they have the same number of cycles of each length.) It follows from (1) that a 1-dimensional representation of S n is completely determined by its values on the transpositions. It follows from (2) that the representation is constant on the transpositions. Finally, since each transposition τ satisfies τ 2 = 1 it follows that this constant value is ±1. Thus there can only be two 1-dimensional representations of S n ; the first takes the value 1 on the transpositions, and so is 1 everywhere; the second takes the value -1 on the transpositions, and takes the value ( 1) r on the permutation g = τ 1 τ r. Thus S n has just two 1-dimensional representations; the trivial representation 1 and the parity representation ɛ. 5. Let s look again at the dihedral group D 4, ie the symmetry group of the square ABCD. Let r denote the rotation through π/2, taking A into B; and let c denote the reflection in AC. It is readily verified that r and c generate D 4, ie each element g D 4 is expressible as a word in r and c (eg g = r 2 cr). This follows for example from Lagrange s Theorem. The subgroup generated by r and c contains at least the 5 elements 1, r, r 2, r 3, c, and so must be the whole group. (We shall sometimes denote the identity element in a group by 1, while at other times we shall use e or I.) It is also easy to see that r and c satisfy the relations r 4 = 1, c 2 = 1, rc = cr 3. In fact these are defining relations for D 4, ie every relation between r and c can be derived from these 3. We can express this in the form D 4 = r, c : r 4 = c 2 = 1, rc = cr 3.

9 424 I 1 9 Now suppose α is a 1-dimensional representation of D 4. Then we must have α(r) 4 = α(c) 2 = 1, α(r)α(c) = α(c)α(r) 3. From the last relation Thus there are just 4 possibilities α(r) 2 = 1. α(r) = ±1, α(c) = ±1. It is readily verified that all 4 of these satisfy the 3 defining relations for s and t. It follows that each defines a homomorphism α : D 4 k. We conclude that D 4 has just 4 1-dimensional representations. 6. We look now at some examples from chemistry and physics. It should be emphasized, firstly that the theory is completely independent of these examples, which can safely be ignored; and secondly, that we are not on oath when speaking of physics. It would be inappropriate to delve too deeply here into the physical basis for the examples we give. Figure 1.2: The methane molecule First let us look at the methane molecule CH 4. In its stable state the 4 hydrogen atoms are situated at the vertices of a regular tetrahedron, with the single carbon atom at its centroid (Figure??). The molecule evidently has symmetry group S 4, being invariant under permutations of the 4 hydrogen atoms. Now suppose the molecule is vibrating about this stable position. We suppose that the carbon atom at the centroid remains fixed. (We shall return to this point later.) Thus the configuration of the molecule at any moment is defined by the displacement of the 4 hydrogen atoms, say X i = (x i1, x i2, x i3 ) (i = 1, 2, 3, 4).

10 424 I 1 10 Since the centroid remains fixed, x ij = 0 (j = 1, 2, 3). i This reduces the original 12 degrees of freedom to 9. Now let us assume further that the angular momentum also remains 0, ie the molecule is not slowly rotating. This imposes a further 3 conditions on the x ij, leaving 6 degrees of freedom for the 12 coordinates x ij. Mathematically, the coordinates are constrained to lie in a 6-dimensional space. In other words we can find 6 generalized coordinates q 1,..., q 6 chosen so that q 1 = q 2 = = q 6 = 0 at the point of equilibrium such that each of the x ij is expressible in terms of the q k : x ij = x ij (q 1,..., q 6 ). The motion of the molecule is governed by the Euler-Lagrange equations ( ) d K = V dt q k q k where K is the kinetic energy of the system, and V its potential energy. (These equations were developed for precisely this purpose, to express the motion of a system whose configuration is defined by generalized coordinates.) The kinetic energy of the system is given in terms of the mass m of the hydrogen atom by K = 1 2 m i,j ẋ 2 ij On substituting we see that x ij = x ij q x ij q 6, q 1 q 6 K = K( q 1,..., q 6 ), where K is a positive-definite quadratic form. Although the coefficients of this quadratic form are actually functions of q 1,..., q 6, we may suppose them constant since we are dealing with small vibrations. The potential energy of the system, which we may take to have minimal value 0 at the stable position, is given to second order by some positivedefinite quadratic form Q in the q k : V = Q(q 1,..., q 6 ) +....

11 424 I 1 11 While we could explicitly choose the coordinates q k, and determine the kinetic energy K, the potential energy form Q evidently depends on the forces holding the molecule together. Fortunately, we can say a great deal about the vibrational modes of the molecule without knowing anything about these forces. Since these two forms are positive-definite, we can simultaneously diagonalize them, ie we can find new generalized coordinates z 1,..., z 6 such that K = ż ż 2 6, V = ω 2 1z ω 2 6z 2 6. The Euler-Lagrange equations now give z i = ω 2 i z i (i = 1,..., 6). Thus the motion is made up of 6 independent harmonic oscillations, with frequencies ω 1,..., ω 6. As usual when studying harmonic or wave motion, life is easier if we allow complex solutions (of which the real solutions will be the real part). Each harmonic oscillation then has 1 degree of freedom: z j = C j e iω jt. The set of all solutions of these equations (ie all possible vibrations of the system) thus forms a 6-dimensional solution-space V. So far we have made no use of the S 4 -symmetry of the CH 4 molecule. But now we see that this symmetry group acts on the solution space V, which thus carries a representation, ρ say, of S 4. Explicitly, suppose π S 4 is a permutation of the 4 H atoms. This permutation is implemented by a unique spatial isometry Π. (For example, the permutation (123)(4) is effected by rotation through 1/3 of a revolution about the axis joining the C atom to the 4th H atom.) But now if we apply this isometry Π to any vibration v(t) we obtain a new vibration Πv(t). In this way the permutation π acts on the solution-space V. In general, the symmetry group G of the physical configuration will act on the solution-space V. The fundamental result in the representation theory of a finite group G (as we shall establish) is that every representation ρ of G splits into parts, each

12 424 I 1 12 corresponding to a simple representation of G. Each finite group has a finite number of such simple representations, which thus serve as the atoms out of which every representation of G is constructed. (There is a close analogy with the Fundamental Theorem of Arithmetic, that every natural number is uniquely expressible as a product of primes.) The group S 4 (as we shall find) has 5 simple representations, of dimensions 1, 1, 2, 3, 3. Our 6-dimensional representation must be the sum of some of these. It is not hard to see that there is just one 1-dimensional mode (up to a scalar multiple) corresponding to a pulsing of the molecule in which the 4 H atoms move in and out (in sync ) along the axes joining them to the central C atom. (Recall that S 4 has just two 1-dimensional representations: the trivial representation, under which each permutation leaves everything unchanged, and the parity representation, in which even permutations leave things unchanged, while odd permutations reverse them. In our case, the 4 atoms must move in the same way under the trivial representation, while their motion is reversed under an odd permutation. The latter is impossible. For by considering the odd permutation (12)(3)(4) we deduce that the first atom is moving out while the second moves in; while under the action of the even permutation (12)(34) the first and second atoms must move in and out together.) We conclude (not rigorously, it should be emphasized!) that ρ = 1 + α + β where 1 denotes the trivial representation of S 4, α is the unique 2-dimensional representation, and β is one of the two 3-dimensional representations. Thus without any real work we ve deduced quite a lot about the vibrations of CH 4. Each of these 3 modes has a distinct frequency. To see that, note that our system and in fact any similar non-relativistic system has a time symmetry corresponding to the additive group R. For if (z j (t) : 1 j 6) is one solution then (z j (t + c)) is also a solution for any constant c R. The simple representations of R are just the 1-dimensional representations t e iωt. (We shall see that the simple representations of an abelian group are always 1-dimensional.) In effect, Fourier analysis the splitting of a function

13 424 I 1 13 or motion into parts corresponding to different frequencies is just the representation theory of R. The actions of S 4 and R on the solution space commute, giving a representation of the product group S 4 R. As we shall see, the simple representations of a product group G H arise from simple representations of G and H: ρ = σ τ. In the present case we must have ρ = 1 E(ω 1 ) + α E(ω 2 ) + β E(ω 3 ), where ω 1, ω 2, ω 3 are the frequencies of the 3 modes. If the symmetry is slightly broken, eg by placing the vibrating molecule in a magnetic field, these degenerate frequencies will split, so that 6 frequencies will be seen: ω 1, ω 2, ω 2, ω 3, ω 3, ω 3, where eg ω 2 and ω 2 are close to ω. This is the origin of multiple lines in spectroscopy. The concept of broken symmetry has become one of the corner-stones of mathematical physics. In grand unified theories distinct particles are seen as identical (like our 4 H atoms) under some large symmetry group, whose action is broken in our actual universe. 7. Vibrations of a circular drum. [?]. Consider a circular elastic membrane. The motion of the membrane is determined by the function z(x, y, t) (x 2 + y 2 r 2 ) where z is the height of the point of the drum at position (x, y). It is not hard to establish that under small vibrations this function will satisfy the wave equation ( 2 ) z T x + 2 z = ρ 2 z 2 y 2 t, 2 where T is the tension of the membrane and ρ its mass per unit area. This may be written ( 2 ) z x + 2 z = 1 2 z 2 y 2 c 2 t, 2 where c = (T/ρ) 1/2 is the speed of the wave motion. The configuration has O(2) symmetry, where O(2) is the group of 2-dimensional isometries leaving the centre O fixed, consisting of the rotations about O and the reflections in lines through O.

14 424 I 1 14 Although this group is not finite, it is compact. As we shall see, the representation theory of compact groups is essentially identical to the finite theory; the main difference being that a compact group has a countable infinity of simple representations. For example, the group O(2) has the trivial representation 1, and an infinity of representations R(1), R(2),..., each of dimension 2. The circular drum has corresponding modes M(0), M(1), M(2),..., each with its characteristic frequency. As in our last example, after taking time symmetry into account, the solution-space carries a representation ρ of the product group O(2) R, which splits into 1 E(ω 0 ) + R(1) E(ω 1 ) + R(2) E(ω 2 ) In the last example but one, we considered the 4 hydrogen atoms in the methane molecule as particles, or solid balls. But now let us consider a single hydrogen atom, consisting of an electron moving in the field of a massive central proton. According to classical non-relativistic quantum mechanics [?], the state of the electron (and so of the atom) is determined by a wave function ψ(x, y, z, t), whose evolution is determined by Schrödinger s equation i ψ t = Hψ. Here H is the hamiltonian operator, given by Hψ = 2 2m 2 ψ + V (r)ψ, where m is the mass of the electron, V (t) is its potential energy, is Planck s constant, and 2 ψ = 2 ψ x ψ y ψ z 2. Thus Schrodinger s equation reads, in full, i ψ ( t = 2 2 ) ψ 2m x + 2 ψ 2 y + 2 ψ e2 2 z 2 r ψ. The essential point is that this is a linear differential equation, whose solutions therefore form a vector space, the solution-space.

15 424 I 1 15 We regard the central proton as fixed at O. (A more accurate account might take O to be the centre of mass of the system.) The system is invariant under the orthogonal group O(3), consisting of all isometries that is, distancepreserving transformations which leave O fixed. Thus the solution space carries a representation of the compact group O(3). This group is a product-group: O(3) = SO(3) C 2, where C 2 = {I, J} (J denoting reflection in O), while SO(3) is the subgroup of orientation-preserving isometries. In fact, each such isometry is a rotation about some axis, so SO(3) is group of rotations in 3 dimensions. The rotation group SO(3) has simple representations D 0, D 1, D 2,... of dimensions 1, 3, 5,.... To each of these corresponds a mode of the hydrogen atom, with a particular frequency ω and corresponding energy level E = ω. These energy levels are seen in the spectroscope, although the spectral lines of hydrogen actually correspond to differences between energy levels, since they arise from photons given off when the energy level changes. This idea considering the space of atomic wave functions as a representation of SO(3) gave the first explanation of the periodic table of the elements, proposed many years before by Mendeleev on purely empirical grounds [?]. The discussion above ignores the spin of the electron. In fact representation theory hints strongly at the existence of spin, since the double-covering SU(2) of SO(3) adds the spin representations D 1/2, D 3/2,... of dimensions 2, 4,... to the sequence above, as we shall see. Finally, it is worth noting that quantum theory (as also electrodynamics) are linear theories, where the Principle of Superposition rules. Thus the application of representation theory is exact, and not an approximation restricted to small vibrations, as in classical mechanical systems like the methane molecule, or the drum. 9. The classification of elementary particles. [?]. Consider an elementary particle E, eg an electron, in relativistic quantum theory. The possible states of E again correspond to the points of a vector space V. More precisely, they correspond to the points of the projective space P (V ) formed by the rays, or 1-dimensional subspaces, of V. For the wave functions ψ and ρψ correspond to the same state of E.

16 424 I 1 16 The state space V is now acted on by the Poincaré group E(1, 3) formed by the isometries of Minkowski space-time. It follows that V carries a representation of E(1, 3). Each elementary particle corresponds to a simple representation of the Poincaré group E(1, 3). This group is not compact. It is however a Lie group; and as we shall see a different approach to representation theory, based on Lie algebras, allows much of the theory to be extended to this case. A last remark. One might suppose, from its reliance on linearity, that representation theory would have no rôle to play in curved space-time. But that is far from true. Even if the underlying topological space is curved, the vector and tensor fields on such a space preserve their linear structure. (So one can, for example, superpose vector fields on a sphere.) Thus representation theory can still be applied; and in fact, the so-called gauge theories introduced in the search for a unified theory of everything are of precisely this kind.

17 Bibliography [1] C. A. Coulson. Waves. Oliver and Boyd, [2] Richard Feynman. Lecture Notes on Physics III. Addison-Wesley, 196. [3] D. J. Simms. Lie Groups and Quantum Mechanics. Springer, [4] Hermann Weyl. The Theory of Groups and Quantum Mechanics. Dover, I 1 17

18 BIBLIOGRAPHY 424 I 1 18 Exercises All representations are over C, unless the contrary is stated. In Exercises determine all 1-dimensional representations of the given group. 1 C 2 2 C 3 3 C n 4 D 2 5 D 3 6 D n 7 Q 8 8 A 4 9 A n 10 Z 11 D = r, s : s 2 = 1, rsr = s Suppose G is a group; and suppose g, h G. The element [g, h] = ghg 1 h 1 is called the commutator of g and h. The subgroup G [G, G] is generated by all commutators in G is called the commutator subgroup, or derived group of G. 12 Show that G lies in the kernel of any 1-dimensional representation ρ of G, ie ρ(g) acts trivially if g G. 13 Show that G is a normal subgroup of G, and that G/G is abelian. Show moreover that if K is a normal subgroup of G then G/K is abelian if and only if G K. [In other words, G is the smallest normal subgroup such that G/G is abelian.) 14 Show that the 1-dimensional representations of G form an abelian group G under multiplication. [Nb: this notation G is normally only used when G is abelian.] 15 Show that C n = C n. 16 Show that for any 2 groups G, H (G H) = G H. 17 By using the Structure Theorem on Finite Abelian Groups (stating that each such group is expressible as a product of cyclic groups) or otherwise, show that A = A for any finite abelian group A. 18 Suppose Θ : G H is a homomorphism of groups. Then each representation α of H defines a representation Θα of G. 19 Show that the 1-dimensional representations of G and of G/G are in oneone correspondence. In Exercises determine the derived group G of the given group G. 20 C n 21 D n 22 Z 23 D 24 Q 8 25 S n 26 A 4 27 A n

19 Chapter 2 Equivalent Representations Every mathematical theory starts from some notion of equivalence an agreement not to distinguish between objects that look the same in some sense. Definition 2.1 Suppose α, β are two representations of G in the vector spaces U, V over k. We say that α and β are equivalent, and we write α = β, if U and V are isomorphic G-spaces. In other words, we can find a linear map which preserves the action of G, ie Remarks: t : U V t(gu) = g(tu) for all g G, u U. 1. Suppose α and β are given in matrix form: α : g A(g), β : g B(g). If α = β, then U and V are isomorphic, and so in particular dim α = dim β, ie the matrices A(g) and B(g) are of the same size. Suppose the linear map t : U V is given by the matrix P. Then the condition t(gu) = g(tu) gives B(g) = P A(g)P 1 for each g G. This is the condition in matrix terms for two representations to be equivalent. 424 I 2 1

20 424 I Recall that two n n matrices S, T are said to be similar if there exists a non-singular (invertible) matrix P such that T = P SP 1. A necessary condition for this is that A, B have the same eigenvalues. For the characteristic equations of two similar matrices are identical: det ( P SP 1 λi ) = det P det(s λi) det P 1 = det(s λi). 3. In general this condition is necessary but not sufficient. For example, the matrices ( ) ( ) , have the same eigenvalues 1,1, but are not similar. (No matrix is similar to the identity matrix I except I itself.) However, there is one important case, or particular relevance to us, where the converse is true. Let us recall a result from linear algebra. An n n complex matrix A is diagonalisable if and only if it satisfies a separable polynomial equation, ie one without repeated roots. It is easy to see that if A is diagonalisable then it satisfies a separable equation. For if λ 1... A λ 1 λ 2... then A satisfies the separable equation m(x) (x λ 1 )(x λ 2 ) = 0. The converse is less obvious. Suppose A satisfies the polynomial equation p(x) (x λ 1 ) (x λ r ) = 0 with λ 1,..., λ r distinct. Consider the expression of 1/p(x) as a sum of partial fractions: 1 p(x) = a a r. x λ 1 x λ r

21 424 I 2 3 Multiplying across, 1 = a 1 Q 1 (x) + + a r Q r (x), where Q i (x) = (x λ j ) = p(x). j i x λ i Substituting x = A, I = a 1 Q 1 (A) + + a r Q r (A). Applying each side to the vector v V, v = a 1 Q 1 (A)v + + a r Q r (A)v = v v r, say. The vector v i is an eigenvector of A with eigenvalue λ i, since (A λ i )v i = a i p(a)v = 0. Thus every vector is expressible as a sum of eigenvectors. In other words the eigenvectors of A span the space. But that is precisely the condition for A to be diagonalisable. For we can find a basis for V consisting of eigenvectors, and with respect to this basis A will be diagonal. 4. It is important to note that while each matrix A(g) is diagonalisable separately, we cannot in general diagonalise all the A(g) simultaneously. That would imply that the A(g) commuted, which is certainly not the case in general. 5. However, we can show that if A 1, A 2,... is a set of commuting matrices then they can be diagonalised simultaneously if and only if they can be diagonalised separately. To see this, suppose λ is an eigenvalue of A 1. Let E = {v : A 1 v = λv} be the corresponding eigenspace. Then E is stable under all the A i, since v E = A 1 (A i v) = A i A 1 v = λa i v = A i v E.

22 424 I 2 4 Thus we have reduced the problem to the simultaneous diagonalisation of the restrictions of A 2, A 3,... to the eigenspaces of A 1. A simple inductive argument on the degree of the A i yields the result. In our case, this means that we can diagonalise some (or all) of our representation matrices A(g 1 ), A(g 2 ),... if and only it these matrices commute. This is perhaps best seen as a result on the representations of abelian groups, which we shall meet later. 6. To summarise, two representations α, β are certainly not equivalent if A(g), B(g) have different eigenvalues for some g G. Suppose to the contrary that A(g), B(g) have the same eigenvalues for all g G. Then as we have seen for all g, ie for some invertible matrix P (g). A(g) B(g) B(g) = P (g)a(g)p (g) 1 Remarkably, we shall see that if this is so for all g G, then in fact α and β are equivalent. In other words, if such a matrix P (g) exists for all g then we can find a matrix P independent of g such that for all g G. 7. Suppose A B, ie B(g) = P A(g)P 1 B = P AP 1. We can interpret this as meaning that A and B represent the same linear transformation, under the change of basis defined by P. Thus we can think of two equivalent representations as being, if effect, the same representation looked at from two points of view, that is, taking two different bases for the representation-space. Example: Let us look again at the natural 2-dimensional real representation ρ of the symmetry group D 4 of the square ABCD. Recall that when we took coordinates with respect to axes Ox, Oy bisecting DA, AB, ρ took the matrix form s ( ) c ( ),

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

Summary of week 8 (Lectures 22, 23 and 24)

WEEK 8 Summary of week 8 (Lectures 22, 23 and 24) This week we completed our discussion of Chapter 5 of [VST] Recall that if V and W are inner product spaces then a linear map T : V W is called an isometry

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

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,

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize

Inner 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,

Elements of Abstract Group Theory

Chapter 2 Elements of Abstract Group Theory Mathematics is a game played according to certain simple rules with meaningless marks on paper. David Hilbert The importance of symmetry in physics, and for

The determinant of a skew-symmetric matrix is a square. This can be seen in small cases by direct calculation: 0 a. 12 a. a 13 a 24 a 14 a 23 a 14

4 Symplectic groups In this and the next two sections, we begin the study of the groups preserving reflexive sesquilinear forms or quadratic forms. We begin with the symplectic groups, associated with

IRREDUCIBLE 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

PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.

PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include

Notes 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

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

Chapter 7. Permutation Groups

Chapter 7 Permutation Groups () We started the study of groups by considering planar isometries In the previous chapter, we learnt that finite groups of planar isometries can only be cyclic or dihedral

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

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

MATH 240 Fall, Chapter 1: Linear Equations and Matrices

MATH 240 Fall, 2007 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 9th Ed. written by Prof. J. Beachy Sections 1.1 1.5, 2.1 2.3, 4.2 4.9, 3.1 3.5, 5.3 5.5, 6.1 6.3, 6.5, 7.1 7.3 DEFINITIONS

Similarity 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

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

Recall 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

Chapter 8. Matrices II: inverses. 8.1 What is an inverse?

Chapter 8 Matrices II: inverses We have learnt how to add subtract and multiply matrices but we have not defined division. The reason is that in general it cannot always be defined. In this chapter, we

CLASSIFYING FINITE SUBGROUPS OF SO 3

CLASSIFYING FINITE SUBGROUPS OF SO 3 HANNAH MARK Abstract. The goal of this paper is to prove that all finite subgroups of SO 3 are isomorphic to either a cyclic group, a dihedral group, or the rotational

Quantum Mechanics and Representation Theory

Quantum Mechanics and Representation Theory Peter Woit Columbia University Texas Tech, November 21 2013 Peter Woit (Columbia University) Quantum Mechanics and Representation Theory November 2013 1 / 30

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

ISOMETRIES 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

(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.

Matrix Algebra LECTURE 1. Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 = a 11 x 1 + a 12 x 2 + +a 1n x n,

LECTURE 1 Matrix Algebra Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 a 11 x 1 + a 12 x 2 + +a 1n x n, (1) y 2 a 21 x 1 + a 22 x 2 + +a 2n x n, y m a m1 x 1 +a m2 x

RESULTANT AND DISCRIMINANT OF POLYNOMIALS

RESULTANT AND DISCRIMINANT OF POLYNOMIALS SVANTE JANSON Abstract. This is a collection of classical results about resultants and discriminants for polynomials, compiled mainly for my own use. All results

MATRIX 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 +

BILINEAR FORMS KEITH CONRAD

BILINEAR FORMS KEITH CONRAD The geometry of R n is controlled algebraically by the dot product. We will abstract the dot product on R n to a bilinear form on a vector space and study algebraic and geometric

1D 3D 1D 3D. is called eigenstate or state function. When an operator act on a state, it can be written as

Chapter 3 (Lecture 4-5) Postulates of Quantum Mechanics Now we turn to an application of the preceding material, and move into the foundations of quantum mechanics. Quantum mechanics is based on a series

Linear 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

A matrix over a field F is a rectangular array of elements from F. The symbol

Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F) denotes the collection of all m n matrices over F Matrices will usually be denoted

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

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

13 Solutions for Section 6

13 Solutions for Section 6 Exercise 6.2 Draw up the group table for S 3. List, giving each as a product of disjoint cycles, all the permutations in S 4. Determine the order of each element of S 4. Solution

Factoring Polynomials

Factoring Polynomials Sue Geller June 19, 2006 Factoring polynomials over the rational numbers, real numbers, and complex numbers has long been a standard topic of high school algebra. With the advent

Practice Math 110 Final. Instructions: Work all of problems 1 through 5, and work any 5 of problems 10 through 16.

Practice Math 110 Final Instructions: Work all of problems 1 through 5, and work any 5 of problems 10 through 16. 1. Let A = 3 1 1 3 3 2. 6 6 5 a. Use Gauss elimination to reduce A to an upper triangular

WHICH LINEAR-FRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE?

WHICH LINEAR-FRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE? JOEL H. SHAPIRO Abstract. These notes supplement the discussion of linear fractional mappings presented in a beginning graduate course

D-MATH Algebra I HS 2013 Prof. Brent Doran. Solution 5

D-MATH Algebra I HS 2013 Prof. Brent Doran Solution 5 Dihedral groups, permutation groups, discrete subgroups of M 2, group actions 1. Write an explicit embedding of the dihedral group D n into the symmetric

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

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

Notes on Orthogonal and Symmetric Matrices MENU, Winter 2013

Notes on Orthogonal and Symmetric Matrices MENU, Winter 201 These notes summarize the main properties and uses of orthogonal and symmetric matrices. We covered quite a bit of material regarding these topics,

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.

CONTROLLABILITY. Chapter 2. 2.1 Reachable Set and Controllability. Suppose we have a linear system described by the state equation

Chapter 2 CONTROLLABILITY 2 Reachable Set and Controllability Suppose we have a linear system described by the state equation ẋ Ax + Bu (2) x() x Consider the following problem For a given vector x in

MATH 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

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

Notes on Factoring. MA 206 Kurt Bryan

The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor

9 MATRICES AND TRANSFORMATIONS

9 MATRICES AND TRANSFORMATIONS Chapter 9 Matrices and Transformations Objectives After studying this chapter you should be able to handle matrix (and vector) algebra with confidence, and understand the

Chapter 7: Products and quotients

Chapter 7: Products and quotients Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 42, Spring 24 M. Macauley (Clemson) Chapter 7: Products

LINEAR ALGEBRA. September 23, 2010

LINEAR ALGEBRA September 3, 00 Contents 0. LU-decomposition.................................... 0. Inverses and Transposes................................. 0.3 Column Spaces and NullSpaces.............................

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.

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

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation

4. FIRST STEPS IN THE THEORY 4.1. A

4. FIRST STEPS IN THE THEORY 4.1. A Catalogue of All Groups: The Impossible Dream The fundamental problem of group theory is to systematically explore the landscape and to chart what lies out there. We

THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS

THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear

GENERATING SETS KEITH CONRAD

GENERATING SETS KEITH CONRAD 1 Introduction In R n, every vector can be written as a unique linear combination of the standard basis e 1,, e n A notion weaker than a basis is a spanning set: a set of vectors

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a

1 Symmetries of regular polyhedra

1230, notes 5 1 Symmetries of regular polyhedra Symmetry groups Recall: Group axioms: Suppose that (G, ) is a group and a, b, c are elements of G. Then (i) a b G (ii) (a b) c = a (b c) (iii) There is an

CHAPTER 12 MOLECULAR SYMMETRY

CHAPTER 12 MOLECULAR SYMMETRY In many cases, the symmetry of a molecule provides a great deal of information about its quantum states, even without a detailed solution of the Schrödinger equation. A geometrical

Classification of Cartan matrices

Chapter 7 Classification of Cartan matrices In this chapter we describe a classification of generalised Cartan matrices This classification can be compared as the rough classification of varieties in terms

The 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

The Mathematics of Origami

The Mathematics of Origami Sheri Yin June 3, 2009 1 Contents 1 Introduction 3 2 Some Basics in Abstract Algebra 4 2.1 Groups................................. 4 2.2 Ring..................................

MATH36001 Background Material 2015

MATH3600 Background Material 205 Matrix Algebra Matrices and Vectors An ordered array of mn elements a ij (i =,, m; j =,, n) written in the form a a 2 a n A = a 2 a 22 a 2n a m a m2 a mn is said to be

ABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS

ABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS John A. Beachy Northern Illinois University 2014 ii J.A.Beachy This is a supplement to Abstract Algebra, Third Edition by John A. Beachy and William D. Blair

26. Determinants I. 1. Prehistory

26. Determinants I 26.1 Prehistory 26.2 Definitions 26.3 Uniqueness and other properties 26.4 Existence Both as a careful review of a more pedestrian viewpoint, and as a transition to a coordinate-independent

x1 x 2 x 3 y 1 y 2 y 3 x 1 y 2 x 2 y 1 0.

Cross product 1 Chapter 7 Cross product We are getting ready to study integration in several variables. Until now we have been doing only differential calculus. One outcome of this study will be our ability

Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

Linear 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

15.062 Data Mining: Algorithms and Applications Matrix Math Review

.6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop

Chapter 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

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

3 1. Note that all cubes solve it; therefore, there are no more

Math 13 Problem set 5 Artin 11.4.7 Factor the following polynomials into irreducible factors in Q[x]: (a) x 3 3x (b) x 3 3x + (c) x 9 6x 6 + 9x 3 3 Solution: The first two polynomials are cubics, so if

UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure

UNIT 2 MATRICES - I Matrices - I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress

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

2.1: MATRIX OPERATIONS

.: MATRIX OPERATIONS What are diagonal entries and the main diagonal of a matrix? What is a diagonal matrix? When are matrices equal? Scalar Multiplication 45 Matrix Addition Theorem (pg 0) Let A, B, and

= 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that

Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without

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

Continuous Groups, Lie Groups, and Lie Algebras

Chapter 7 Continuous Groups, Lie Groups, and Lie Algebras Zeno was concerned with three problems... These are the problem of the infinitesimal, the infinite, and continuity... Bertrand Russell The groups

1 Spherical Kinematics

ME 115(a): Notes on Rotations 1 Spherical Kinematics Motions of a 3-dimensional rigid body where one point of the body remains fixed are termed spherical motions. A spherical displacement is a rigid body

Facts About Eigenvalues By Dr David Butler Definitions Suppose A is an n n matrix An eigenvalue of A is a number λ such that Av = λv for some nonzero vector v An eigenvector of A is a nonzero vector v

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

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

Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain

Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain 1. Orthogonal matrices and orthonormal sets An n n real-valued matrix A is said to be an orthogonal

MATH2001 Development of Mathematical Ideas History of Solving Polynomial Equations

MATH2001 Development of Mathematical Ideas History of Solving Polynomial Equations 19/24 April 2012 Lagrange s work on general solution formulae for polynomial equations The formulae for the cubic and

a 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

Theorem 5. The composition of any two symmetries in a point is a translation. More precisely, S B S A = T 2

Isometries. Congruence mappings as isometries. The notion of isometry is a general notion commonly accepted in mathematics. It means a mapping which preserves distances. The word metric is a synonym to

PROVING STATEMENTS IN LINEAR ALGEBRA

Mathematics V2010y Linear Algebra Spring 2007 PROVING STATEMENTS IN LINEAR ALGEBRA Linear algebra is different from calculus: you cannot understand it properly without some simple proofs. Knowing statements

Section 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

Math 312 Homework 1 Solutions

Math 31 Homework 1 Solutions Last modified: July 15, 01 This homework is due on Thursday, July 1th, 01 at 1:10pm Please turn it in during class, or in my mailbox in the main math office (next to 4W1) Please

Additional Topics in Linear Algebra Supplementary Material for Math 540. Joseph H. Silverman

Additional Topics in Linear Algebra Supplementary Material for Math 540 Joseph H Silverman E-mail address: jhs@mathbrownedu Mathematics Department, Box 1917 Brown University, Providence, RI 02912 USA Contents

BANACH 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

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

1 Eigenvalues and Eigenvectors

Math 20 Chapter 5 Eigenvalues and Eigenvectors Eigenvalues and Eigenvectors. Definition: A scalar λ is called an eigenvalue of the n n matrix A is there is a nontrivial solution x of Ax = λx. Such an x

Recall the basic property of the transpose (for any A): v A t Aw = v w, v, w R n.

ORTHOGONAL MATRICES Informally, an orthogonal n n matrix is the n-dimensional analogue of the rotation matrices R θ in R 2. When does a linear transformation of R 3 (or R n ) deserve to be called a rotation?

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

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,

Diagonalisation. Chapter 3. Introduction. Eigenvalues and eigenvectors. Reading. Definitions

Chapter 3 Diagonalisation Eigenvalues and eigenvectors, diagonalisation of a matrix, orthogonal diagonalisation fo symmetric matrices Reading As in the previous chapter, there is no specific essential

THREE DIMENSIONAL GEOMETRY

Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,