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Linear Algebra & Geometry Linear Algebra & Geometry Roman Schubert1 May 22, 2012 1School of Mathematics, University of Bristol. c University of Bristol 2011 This material is copyright of the University unless explicitly stated oth- erwise. It is provided exclusively for educational purposes at the University and is to be downloaded or copied for your private study only. 2 These are Lecture Notes for the 1st year Linear Algebra and Geometry course in Bristol. This is an evolving version of them, and it is very likely that they still contain many misprints. Please report serious errors you find to me ([email protected]) and I will post an update on the Blackboard page of the course. These notes cover the main material we will develop in the course, and they are meant to be used parallel to the lectures. The lectures will follow roughly the content of the notes, but sometimes in a different order and sometimes containing additional material. On the other hand, we sometimes refer in the lectures to additional material which is covered in the notes. Besides the lectures and the lecture notes, the homework on the problem sheets is the third main ingredient in the course. Solving problems is the most efficient way of learning mathematics, and experience shows that students who regularly hand in homework do reasonably well in the exams. These lecture notes do not replace a proper textbook in Linear Algebra. Since Linear Algebra appears in almost every area in Mathematics a slightly more advanced textbook which complements the lecture notes will be a good companion throughout your mathematics courses. There is a wide choice of books in the library you can consult. 1 1 c University of Bristol 2013 This material is copyright of the University unless explicitly stated otherwise. It is provided exclusively for educational purposes at the University and is to be downloaded or copied for your private study only. Contents 1 The Euclidean plane and complex numbers 5 1.1 The Euclidean plane R2 .............................. 5 1.2 The dot product and angles . 9 1.3 Complex Numbers . 10 2 Euclidean space Rn 15 2.1 Dot product . 17 2.2 Angle between vectors in Rn ........................... 19 2.3 Linear subspaces . 20 3 Linear equations and Matrices 25 3.1 Matrices . 29 3.2 The structure of the set of solutions to a system of linear equations . 33 3.3 Solving systems of linear equations . 36 3.3.1 Elementary row operations . 36 3.4 Elementary matrices and inverting a matrix . 41 4 Linear independence, bases and dimension 45 4.1 Linear dependence and independence . 45 4.2 Bases and dimension . 48 4.3 Orthonormal Bases . 52 5 Linear Maps 55 5.1 Abstract properties of linear maps . 57 5.2 Matrices . 60 5.3 Rank and nullity . 62 6 Determinants 65 6.1 Definition and basic properties . 65 6.2 Computing determinants . 74 6.3 Some applications of determinants . 78 6.3.1 Inverse matrices and linear systems of equations . 79 6.3.2 Bases . 80 6.3.3 Cross product . 80 3 4 CONTENTS 7 Vector spaces 83 7.1 On numbers . 83 7.2 Vector spaces . 85 7.3 Subspaces . 89 7.4 Linear maps . 91 7.5 Bases and Dimension . 93 7.6 Direct sums . 98 7.7 The rank nullity Theorem . 100 7.8 Projections . 102 7.9 Isomorphisms . 104 7.10 Change of basis and coordinate change . 105 7.11 Linear maps and matrices . 109 8 Eigenvalues and Eigenvectors 117 9 Inner product spaces 127 10 Linear maps on inner product spaces 137 10.1 Complex inner product spaces . 138 10.2 Real matrices . 144 Chapter 1 The Euclidean plane and complex numbers 1 1.1 The Euclidean plane R2 To develop some familiarity with the basic concepts in linear algebra let us start by discussing the Euclidean plane R2: Definition 1.1. The set R2 consists of ordered pairs (x; y) of real numbers x; y 2 R. Remarks: • In the lecture we will denote elements in R2 often by underlined letters and arrange the numbers x; y vertically x v = y Other common notations for elements in R2 are by boldface letters v, and this is the notation we will use in these notes, or by an arrow above the letter ~v. But often no special notation is used at all and one writes v 2 R2 and v = (x; y). x y • That the pair is ordered means that 6= if x 6= y. y x • The two numbers x and y are called the x-component, or first component, and the y-component, or second component, respectively. For instance the vector 1 2 has x-component 1 and y-component 2. • We visualise a vector in R2 as a point in the plane, with the x-component on the horizontal axis and the y-component on the vertical axis. 5 6 CHAPTER 1. THE EUCLIDEAN PLANE AND COMPLEX NUMBERS x v v −w v+w y w Figure 1.1: Left: An element v = (y; x) in R2 represented by a vector in the plane. Right: vector addition, v + w, and the negative −v. We will define two operations on vectors. The first one is addition: v w Definition 1.2. Let v = 1 ; w = 1 2 R2, then we define the sum of v and w by v2 w2 v + w v + w := 1 1 v2 + w2 And the second operation is multiplication by real numbers: v Definition 1.3. Let v = 1 2 R2 and λ 2 R, then we define the product of v by λ by v2 λv λv := 1 λv2 Some typical quantities in nature which are described by vectors are velocities and forces. The addition of vectors appears naturally for these, for example if a ship moves through the water with velocity vS and there is a current in the water with velocity vC , then the velocity of the ship over ground is vS + vC . By combining these two relation we can form expressions like λv + µw for v; w 2 R2 and λ, µ 2 R, we call this a linear combination of v and w. For instance 1 0 5 0 5 5 + 6 = + = : −1 2 −5 12 7 2 We can as well consider linear combinations of k vectors v1; v2; ··· ; vk 2 R with coefficients λ1; λ2 ··· ; λk 2 R, k X λ1v1 + λ2v2 + ··· + λkvk = λivi i=1 1 c University of Bristol 2011 This material is copyright of the University unless explicitly stated otherwise. It is provided exclusively for educational purposes at the University and is to be downloaded or copied for your private study only. 1.1. THE EUCLIDEAN PLANE R2 7 0 Notice that 0v = for any v 2 R2 and we will in the following denote the vector 0 whose entries are both 0 by 0, so we have v + 0 = 0 + v = v for any v 2 R2. We will use as well the shorthand −v to denote (−1)v and w−v := w+(−1)v. Notice that with this notation v − v = 0 for all v 2 R2. The norm of a vector is defined by v Definition 1.4. Let v = 1 2 R2, then the norm of v is defined by v2 q 2 2 kvk := v1 + v2 : By Pythagoras Theorem the norm is just the geometric length of the distance between the point in the plane with coordinates (v1; v2) and the origin 0. Furthermore kv − wk is the distance between the points v and w. 5 For instance the norm of a vector of the form v = , which has no y component, is 0 3 p p just kvk = 5, whereas if w = we find kwk = 9 + 1 = 10 and the distance between −1 p p v and w is kv − wk = 4 + 1 = 5. Let us now look how the norm relates to the structures we defined previously, namely addition and scalar multiplication: Theorem 1.5. The norm satisfies (i) kvk ≥ 0 for all v 2 R2 and kvk = 0 if and only if v = 0. (ii) kλvk = jλjkvk for all λ 2 R; v 2 R2 (iii) kv + wk ≤ kvk + kwk for all v; w 2 R2. Proof. We will only prove the first two statements, the third statement, which is called the triangle inequality will be proved in the exercises. p 2 2 For the first statement we use the definition of the norm kvk = v1 + v2 ≥ 0. It is clear 2 2 that k0k = 0, but if kvk = 0, then v1 +v2 = 0, but this is a sum of two non-negative numbers, so in order that they add up to 0 they must both be 0, hence v1 = v2 = 0 and so v = 0 The second statement follows from a direct computation: q p q p 2 2 2 2 2 2 2 2 kλvk = (λv1) + (λv2) = λ (v1 + v2) = λ v1 + v2 = jλjkvk : We have represented a vector by its two components and interpreted them as Cartesian coordinates of a point in R2. We could specify a point in R2 as well by giving its distance λ to the origin and the angle between the line connecting the point to the origin and the x-axis. We will develop this idea, which leads to polar coordinates in calculus, a bit more: 8 CHAPTER 1. THE EUCLIDEAN PLANE AND COMPLEX NUMBERS Definition 1.6. A vector u 2 R2 is called a unit vector if kuk = 1. Remark: A unit vector has length one, hence all unit vectors lie on the circle of radius one in R2, therefore a unit vector is determined by its angle θ with the x-axis.
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