# curs algebra liniara

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• Lecture Noteson

Linear Algebra,Analytic and Dierential Geometry

Nicoleta Brnzei

Jan. 2008

• ii

• Contents

To Students v

1 Linear Algebra 11.1 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Denition and Examples . . . . . . . . . . . . . . . . . . . 11.1.2 Subspaces in a Vector Space . . . . . . . . . . . . . . . . . 3

1.2 Basis and Dimension of a Vector Space . . . . . . . . . . . . . . . 61.2.1 Subspace Spanned by a Subset of a Vector Space . . . . . 61.2.2 Linear Dependence and Independence . . . . . . . . . . . 61.2.3 Basis and Dimension . . . . . . . . . . . . . . . . . . . . . 81.2.4 Changes of Bases . . . . . . . . . . . . . . . . . . . . . . . 10

1.3 Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . 121.3.1 Denition and Examples . . . . . . . . . . . . . . . . . . . 121.3.2 Kernel and Image of a Linear Transformation . . . . . . . 121.3.3 Linear Transformations on Finite-dimensional Vector Spaces 141.3.4 All Vector Spaces of Dimension n over the Same Field Are

Isomorphic . . . . . . . . . . . . . . . . . . . . . . . . . . 151.4 Eigenvectors and Eigenvalues . . . . . . . . . . . . . . . . . . . . 161.5 Quadratic forms; Canonical Form . . . . . . . . . . . . . . . . . . 20

2 Analytic Geometry 232.1 Free (Spatial) Vectors . . . . . . . . . . . . . . . . . . . . . . . . 23

2.1.1 The Vector Space of Free Vectors . . . . . . . . . . . . . . 232.1.2 Dot (Scalar) Product of Free Vectors . . . . . . . . . . . . 262.1.3 The Cross (Vector) Product of Two Vectors . . . . . . . . 272.1.4 Mixed Triple Product . . . . . . . . . . . . . . . . . . . . 28

2.2 Equation of Lines and Planes in Space . . . . . . . . . . . . . . . 292.2.1 The Plane Containing a Point and Two Directions . . . . 292.2.2 The Plane through a Point, Having a Given Normal Di-

rection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.2.3 Equations of a Straight Line in Space . . . . . . . . . . . 32

2.3 Angles and Distances in Space . . . . . . . . . . . . . . . . . . . 342.3.1 Angles: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.3.2 Distances in Space . . . . . . . . . . . . . . . . . . . . . . 35

iii

• iv PREFACE

2.4 Coordinate Transformations . . . . . . . . . . . . . . . . . . . . . 372.4.1 Translations in Plane and in Space . . . . . . . . . . . . . 372.4.2 Changes of Orthonormal Bases in Plane and in Space . . 38

2.5 Conic Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.5.1 Denition and Examples . . . . . . . . . . . . . . . . . . . 412.5.2 The Standard Form of a Conic . . . . . . . . . . . . . . . 442.5.3 Center, Axes and Asymptotes of a Conic . . . . . . . . . 482.5.4 Alternative Denitions of Conics. Eccentricity . . . . . . 50

2.6 Quadrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.6.1 The Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . 512.6.2 Reduced Canonical Equations of Other Quadrics . . . . . 53

2.7 Generated Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 582.7.1 Cylindrical Surfaces . . . . . . . . . . . . . . . . . . . . . 592.7.2 Conic Surfaces . . . . . . . . . . . . . . . . . . . . . . . . 612.7.3 Conoidal Surfaces with a Directing Plane . . . . . . . . . 632.7.4 Surfaces of Revolution . . . . . . . . . . . . . . . . . . . . 64

3 Dierential Geometry 673.1 Plane Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.1.1 Arc Length of a Plane Curve . . . . . . . . . . . . . . . . 683.1.2 Contact Between Two Intersecting Curves . . . . . . . . . 713.1.3 Tangent and Normal Line at a Regular Point . . . . . . . 723.1.4 Osculating Circle; Curvature and Radius of Curvature . . 74

3.2 Space Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773.2.1 Representations . . . . . . . . . . . . . . . . . . . . . . . . 773.2.2 Arc Length of a Space Curve . . . . . . . . . . . . . . . . 783.2.3 The TNB Frame (The Frenet Frame) . . . . . . . . . . . 793.2.4 Curvature and Torsion . . . . . . . . . . . . . . . . . . . . 83

3.3 Dierential Geometry of Surfaces . . . . . . . . . . . . . . . . . . 843.3.1 Representations . . . . . . . . . . . . . . . . . . . . . . . . 843.3.2 Curves on a Surface . . . . . . . . . . . . . . . . . . . . . 853.3.3 Tangent Plane and Surface Normal at a Regular Point . . 873.3.4 First Fundamental Form. Applications . . . . . . . . . . . 88

• To Students

This text gathers the lecture notes of a 14 weeks course taught by the authorat technical faculties and it is a "user manual" especially designed for thosewho might need in their further work some basic knowledge on Linear Algebraand/or Analytic and Dierential 2D- and 3D- Geometry. So, with several simpleexceptions, I did not include any proofs of theorems. Instead of these, I preferredto insert examples and comments or intuitive interpretations.The reader is supposed to have studied at an acceptable level:

matrices, determinants and linear systems; the equation of a line in plane; partial derivatives.

For a better understanding of the notions and results, I suggest the followingway of passing through every subsection of the course.

1. read carefully the theoretical results, trying to understand them; basically,all the information in the "theory" is needed in solving the exercises.

2. try to do by yourself the solved examples. If you dont succeed, repeatstep 1 and eventually, try to recall the necessary high school Mathematicsnotions (dont hesitate to look them up in books or on the Internet. Asimple Google search could be of real help :-) )

3. Try to solve the exercises inside the subsection.

Feel free to express any comments or questions regarding the contents of thecourse, at: nico.brinzei@rdslink.ro.

I wish you success!Yours,Nicoleta Brnzei

v

• vi TO STUDENTS

• Chapter 1

Linear Algebra

1.1 Vector Spaces

1.1.1 Denition and Examples

Informally speaking, a vector space (or a linear space) is a set of objects (calledvectors) that may be scaled and added according to some rules; these objectscan be numbers, (geometrical) free vectors, matrices, functions etc. The theoryof vector spaces is, in some sense, a "theory of everything", which provides apowerful tool for other branches of Mathematics, such as: Geometry, Analysis,Dierential equations.

Let V 6= ? be a non-empty set, and (K;+; ) a commutative eld. Let usdene two operations:

+ : V V ! V; (u; v)! u+ v;(internal operation on V ) which will be called vector addition and

K : K V ! V; (; v)! v;(external operation) called further, scalar multiplication.

Denition 1.1 (V;+; K) is called a vector space over K if:1. (V;+) is an Abelian group, this is:

(a) V is closed under vector addition: 8 u; v 2 V : u+ v 2 V ;(b) vector addition is associative: for all u; v; w 2 V , we have u + (v +

w) = (u+ v) + w;

(c) vector addition is commutative: 8v; w 2 V : v + w = w + v;(d) vector addition has an identity (a neutral) element: 90 2 V; such that

v + 0 = v, 8v 2 V ;

1

• 2 CHAPTER 1. LINEAR ALGEBRA

(e) any element v 2 V admits a symmetric (or an additive inverse)(v) 2 V such that v + (v) = 0:

2. the scalar multiplication K has the following properties:

(a) V is closed under scalar multiplication: 8 2 K; 8v 2 V : v 2 V ;(b) K is distributive w.r.t. vector addition: 8 2 K; 8u; v 2 V : (u +

v) = u+ v;

(c) K is distributive w.r.t. the addition in K: 8; 2 K; 8v 2 V :(+ )v = v + v;

(d) scalar multiplication is compatible with multiplication in K ("asso-ciativity"): 8; 2 K; 8v 2 V : (v) = ()v;

(e) the neutral (identity) element 1 2 K is an identity element for scalarmultiplication: 8v 2 V : 1v = v:

The elements u; v; ::: 2 V are called vectors, while the elements ; ; ::: 2 Kare called scalars.

Rules of computation in vector spaces:

1. Scalar multiplication with the zero vector 0 2 V yields the zero vector:

8 2 K : 0 = 0:

2. Scalar multiplication by the scalar 0 2 K yields the zero vector:

8v 2 V : 0v = 0:

3. No other scalar multiplication leads to the zero vector: if v = 0; theneither = 0; or v = 0:

4. The result of scalar multiplication (1) is the additive inverse of the vector:

8v 2 V : (1)v = v:

If the eld of scalars K is R; then the vector space is called a real vectorspace; if K = C; then V is called a complex vector space.

Examples of vector spaces

1. Let K = R; and Rn = fx = (x1; x2; :::; xn) j xi 2 R; i = 1; ng; the set ofall ordered n-uples of real numbers, and the operations:

x = (x1; x2; :::; xn) 2 Rny = (y1; y2; :::; yn) 2 Rn ! x+ y = (x1 + y1; x2 + y2; :::; xn + yn)

• 1.1. VECTOR SPACES 3

andx = (x1; x2; :::; xn) 2 Rn ! x = (x1; x2; :::; xn):

Then, (Rn;+; R) is a vector space, called the real coordinate space Rn:This is perhaps the most important example of vector space; especially,the cases n = 2 and n = 3 are important for Geometry.

2. Let K be a eld and Mmn(K) the space of m n-type matrices withentries aij 2 K. Then, (Mmn(K);+; K) is a vector space over K: Inparticular, for K = R; (Mmn(R);+; R) is a real vector space.

3. The set K[X] of all polynomials with coe cients in a eld K is a vectorspace over K:

4. The set F(I) of all real-valued functions dened over an interval I R;F(I) = ff j f : I ! Rg;

with the usual addition and scalar multiplication of functions,

(f + g)(x) = f(x) + g(x);

(f)(x) = f(x); 8x 2 I; 2 R;is a real vector space.

5. ("the last, but not the least"), the set E3 of all free vectors a in space,with the usual addition (e.g., using the parallelogram rule) and scalarmultiplication