Introduction to Linear AlgebraThis is a short text in linear algebra, intended for a one-term course. In the first chapter, Lang discusses the relation between the geometry and the algebra underlying the subject, and gives concrete examples of the notions which appear later in the book. He then starts with a discussion of linear equations, matrices and Gaussian elimination, and proceeds to discuss vector spaces, linear maps, scalar products, determinants, and eigenvalues. The book contains a large number of exercises, some of the routine computational type, while others are conceptual. |
Contents
CHAPTER | 2 |
1 Definition of Points in Space | 4 |
2 Located Vectors | 10 |
3 Scalar Product | 12 |
4 The Norm of a Vector | 15 |
5 Parametric Lines | 30 |
6 Planes | 34 |
CHAPTER II | 42 |
3 Convex Sets | 99 |
4 Linear Independence | 104 |
5 Dimension | 110 |
6 The Rank of a Matrix | 115 |
CHAPTER IV | 123 |
3 The Kernel and Image of a Linear Map | 136 |
5 The Matrix Associated with a Linear Map | 150 |
2 Inverses | 164 |
1 Matrices | 43 |
2 Multiplication of Matrices | 47 |
3 Homogeneous Linear Equations and Elimination | 64 |
4 Row Operations and Gauss Elimination | 70 |
5 Row Operations and Elementary Matrices | 77 |
6 Linear Combinations | 85 |
CHAPTER III | 88 |
2 Linear Combinations | 93 |
Other editions - View all
Common terms and phrases
3-space arbitrary assume b₁ c₁ called Chapter characteristic polynomial coefficients column vectors complex numbers component convex set coordinate vector coordinates D(A¹ define denote Det(A determinant dimension dot product eigenvalues eigenvectors equal Example Exercises exist numbers formula functions give Hence i-th inverse j-th column kernel Let F linear combination linear equations linear map linearly independent linearly independent elements located vector n-tuples non-zero orthogonal basis parallelogram spanned perpendicular plane positive definite positive definite scalar Proof properties prove r-th real numbers row equivalent row rank s-th row satisfies scalar multiples scalar product second row Show solve space of solutions square matrix subspace subtract Suppose symmetric system of linear t₁ Theorem 3.1 third row transpose unit vectors v₂ variables vector space Vol(v w₁ write x₁ zero