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Linear Algebra—A Geometric Introduction I LINEAR ALGEBRA—A GEOMETRIC INTRODUCTION I J. B. Cooper Johannes Kepler Universit¨at Linz Contents 1 LINEAREQUATIONSANDMATRIXTHEORY 4 1.1 Systems of linear equations, Gaußian elimination . 4 1.2 Matrices and their arithmetic . 9 1.3 Matrix multiplication, Gaußian elimination andhermitianforms ....................... 29 1.4 Therankofamatrix ....................... 39 1.5 Matrixequivalence ........................ 45 1.6 Blockmatrices .......................... 50 1.7 Generalisedinverses. .. .. 55 1.8 Circulants,Vandermondematrices. 58 2 ANALYTIC GEOMETRY IN 2 and 3 DIMENSIONS 61 2.1 Basicgeometryintheplane . 61 2.2 Angleandlength ......................... 67 2.3 ThreePropositionsofEuclideangeometry . 71 2.4 Affinetransformations . 75 2.5 Isometries and their classification . 78 2.6 Conicsections........................... 89 2.7 Threedimensionalspace . 94 2.8 Vector products, triple products, 3 3determinants. 99 × 2.9 Covariantandcontravariantvectors . 104 2.10 Isometries of R3 ..........................107 3 VECTOR SPACES 113 3.1 The axiomatic definition, linear dependence andlinearcombinations . .113 3.2 Bases(Steinitz’Lemma) . .117 1 3.3 Complementarysubspaces . .122 3.4 Isomorphisms,transfermatrices . 124 3.5 Affinespaces............................127 4 LINEAR MAPPINGS 132 4.1 Definitionsandexamples . .132 4.2 Linearmappingsandmatrices . .134 4.3 Projections and splittings . 139 4.4 Generalised inverses of linear mappings . 143 4.5 Norms and convergence in vector spaces . 144 2 Preface In these notes we have tried to present the theory of matrices, vector spaces and linear operators in a form which preserves a balance be- tween an approach which is too abstract and one which is merely compu- tational. The typical linear algebra course in the early sixties tended to be very computational in nature, with emphasis on the calculation of various canonical forms, culminating in a treatment of the Jordan form without men- tioning the geometrical background of the constructions. More recently, such courses have tended to be more conceptual in form beginning with abstract vector spaces (or even modules over rings!) and developing the theory in the Bourbaki style. Of course, the advantage of such an approach is that it provides easy access and a very efficient approach to a great body of classical mathematics. Unfortunately, this was not exploited in many such courses, leaving the abstract theory a torso, deprived of any useful sense for many students. The present course is an attempt to combine the advantages of both approaches. It begins with the theory of linear equations. The method of Gaußian elimination is explained and, with this material as motivation, matrices and their arithmetic are introduced. The method of solution and its consequences are re-interpreted in terms of this arithmetic. This is followed by a chapter on analytic geometry in two and three dimensions. Two by two matrices are interpreted as geometric transformations and it is shown how the arithmetic of matrices can be used to obtain significant geometrical results. The material of these first two chapters and this dual interpretation of matrices provides the basis for a meaningful transition to the axiomatic theory of vector spaces and linear mappings. After this introductory material, I have felt free to use increasingly higher levels of abstraction in the remainder of the book which deals with determi- nants, the eigenvalue problem, diagonalisation and the Jordan form, spectral theory and multilinear algebra. I have also included a brief introduction to the complex numbers. The book contains several topics which are not usu- ally covered in introductory text books—of which we mentioned generalised inverses (including Moore-Penrose inverses), singular values and the classi- fication of the isometries in R2 and R3. It is also accompanied by three further volumes, one on elementary geometry (based on affine transforma- tions) one containing a further set of exercises and one an introduction to abstract algebra and number theory. I have taken pains to include a large number of worked examples and ex- ercises in the text. The latter are of two types. Each set begins with routine computational exercises to allow the reader to familiarise himself with the concepts and proofs just covered. These are followed by more theoretical ex- ercises which are intended to serve the dual purpose of providing the student with more challenging problems and of introducing him to the rich array of 3 mathematics which has been made accessible by the theory developed. 1 LINEAR EQUATIONS AND MATRIX THE- ORY 1.1 Systems of linear equations, Gaußian elimination The subject of our first chapter is the classical theory of linear equations and matrices. We begin with the elementary treatment of systems of linear equa- tions. Before attempting to derive a general theory, we consider a concrete example: Examples. Solve the system 3x y =6 (1) − x + 3y 2z =1 (2) 2x + 2y −+ 2z = 2. (3) We use the familiar method of successively eliminating the variables: re- placing (2) by 3 (2) (1) and (3) by 3 (3) 2(1) we get the system: · − · − 3x y = 6 (4) − 10y 6z = 3 (5) 8y −+ 6z = −6. (6) − We now proceed to eliminate y in the same manner and so reduce to the system: 3x y = 6 (7) − 10y 6z = 3 (8) − 54z = −18 (9) − and this system can be solved “backwards” i.e. by first calculating z from the final equation, then y from (8) and finally x. This gives the solution 1 1 11 z = y = x = . −3 −2 6 (Of course, it would have been more sensible to use (2) in the original system to eliminate x from (1) and (3). We have deliberately chosen this more me- chanical course since the general method we shall proceed to develop cannot take such special features of a concrete system into account). In order to treat general equations we introduce a more efficient notation: a11x1 + ... + a1nxn = y1 . am1x1 + ... + amnxn = ym 4 is the general system of m equations in n unknowns. The aij are the co- efficients and the problem is to determine the values of the x for given y1,...,ym. In this context we can describe the method of solution used above as follows. (For simplicity, we assume that m = n). We begin by subtracting suitable multiples of the first equation from the later ones in such a way that the coefficients of x1 in these equations vanish. We thus obtain a system of the form b11x1 + b12x2 + ... + b1nxn = z1 b22x2 + ... + b2nxn = z2 . bn1x2 + ... + bnnxn = zn. (with new coefficients and a new right hand side). Ignoring the first equation, we now have a reduced system of (n 1) equations in (n 1) unknowns and, − − proceeding in the same way with this system, then with the resulting system of (n 2) equations and so on, we finally arrive at one of the form: − d11x1 + d12x2 + ... + d1nxn = w1 d22x2 + ... + d2nxn = w2 . dnnxn = wn which we can solve simply by calculating the value of xn from the last equa- tion, substituting this in the second last equation and so working backwards to x1. This all sounds very simple but, unfortunately, reality is rather more complicated. If we call the coefficient of xi in the i-th equation after (i 1) steps in the above procedure the i-th pivot element, then the applicability− of our method depends on the fact that the i-th pivot is non-zero. If it is zero, then two things can happen. Case 1: the coefficient of xi in a later equation (say the j-th one) is non-zero. Then we simply exchange the i-th and j-th equations and proceed as before. Case 2: the coefficients of xi in the i-th and all following equations vanish. In this case, it may happen that the system has no solution. Examples. Consider the systems x + y + 2z = 2 2x + 3y z = 1 − − 3x + 3y 4z =6 6x + y + z = 3 2x y +− 6z = 1 4x + 2y = 1. − − − 5 If we apply the above method to the first system we get successively x + y + 2z = 2 − 10z = 12 3y +− 2z = 3 − i.e. x + y + 2z = 2 3y + 2z =− 3 − 10z = 12 − and this is solvable. In the second case, we get: 2x + 3y z = 1 8y −+ 4z = 0 − 4y + 2z = 3. − − At this stage we see at a glance that the system is not solvable (the last two equations are incompatible) and indeed the next step leads to the system 2x + 3y z = 1 8y −+ 4z = 0 − 0 =6 and the third pivot vanishes. Note that the vanishing of a pivot element does not automatically imply the non-solvability of the equation. For example, if the right hand side of our original equation had been 4 8 6 then it would indeed have been solvable as the reader can verify for himself. Hence we see that if a pivot element is zero in the non-trivial manner of case 2 above, then we require a more subtle analysis to decide the solvability or non-solvability of the system. This will be the main concern of this Chapter whereby the method and result will be repeatedly employed in later ones. A useful tool will be the so-called matrix formalism which we introduce in the next section. Examples. Solve the following systems (if possible): 2x + 4y + z = 1 x + 5y + 2z = 9 3x + 5y = 1 x + y + 7z = 6 5x + 13y + 7z = 5 3y + 4z = 2. − − 6 Solution: In the first case we subtract 7 times the first equation from the third one and get 9x 15y = 2 − − − which is incompatible with the second equation. In the second case, we subtract the 2nd one from the first one and get 4y 5z =3. − This, together with the third equation, gives y = 2, z = 1. Hence x = 3 − and this solution is unique. Exercises: 1) Solve the following systems (if possible): y1 + y2 = a y2 + y3 = b y3 + y4 = c y1 + y4 = d.
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