Linear algebra
Janich, Klaus
Linear algebra - New Delhi Springer India Ltd., 2004 - IX+204 p. 23 cm ;Pbk
58 illustrations
1. Sets and Maps --
1.1 Sets --
1.2 Maps --
1.3 Test --
1.4 Remarks on the Literature --
1.5 Exercises --
2. Vector Spaces --
2.1 Real Vector Spaces --
2.2 Complex Numbers and Complex Vector Spaces --
2.3 Vector Subspaces --
2.4 Test --
2.5 Fields --
2.6 What Are Vectors? --
2.7 Complex Numbers 400 Years Ago --
2.8 Remarks on the Literature --
2.9 Exercises --
3. Dimension --
3.1 Linear Independence --
3.2 The Concept of Dimension --
3.3 Test --
3.4 Proof of the Basis Extension Theorem and the Exchange Lemma --
3.5 The Vector Product --
3.6 The 'Steinitz Exchange Theorem' --
3.7 Exercises --
4. Linear Maps --
4.1 Linear Maps --
4.2 Matrices --
4.3 Test --
4.4 Quotient Spaces --
4.5 Rotations and Reflections in the Plane --
4.6 Historical Aside --
4.7 Exercises --
5. Matrix Calculus --
5.1 Multiplication --
5.2 The Rank of a Matrix --
5.3 Elementary Transformations --
5.4 Test --
5.5 How Does One Invert a Matrix? --
5.6 Rotations and Reflections (continued) --
5.7 Historical Aside --
5.8 Exercises --
6. Determinants --
6.1 Determinants --
6.2 Determination of Determinants --
6.3 The Determinant of the Transposed Matrix --
6.4 Determinantal Formula for the Inverse Matrix --
6.5 Determinants and Matrix Products --
6.6 Test --
6.7 Determinant of an Endomorphism --
6.8 The Leibniz Formula --
6.9 Historical Aside --
6.10 Exercises --
7. Systems of Linear Equations --
7.1 Systems of Linear Equations --
7.2 Cramer's Rule --
7.3 Gaussian Elimination --
7.4 Test --
7.5 More on Systems of Linear Equations --
7.6 Captured on Camera! --
7.7 Historical Aside --
7.8 Remarks on the Literature --
7.9 Exercises --
8. Euclidean Vector Spaces --
8.1 Inner Products --
8.2 Orthogonal Vectors --
8.3 Orthogonal Maps --
8.4 Groups --
8.5 Test --
8.6 Remarks on the Literature --
8.7 Exercises --
9. Eigenvalues --
9.1 Eigenvalues and Eigenvectors --
9.2 The Characteristic Polynomial --
9.3 Test --
9.4 Polynomials --
9.5 Exercises --
10. The Principal Axes Transformation --
10.1 Self-Adjoint Endomorphisms --
10.2 Symmetric Matrices --
10.3 The Principal Axes Transformation for Self-Adjoint Endomorphisms --
10.4 Test --
10.5 Exercises --
11. Classification of Matrices --
11.1 What Is Meant by 'Classification'? --
11.2 The Rank Theorem --
11.3 The Jordan Normal Form --
11.4 More on the Principal Axes Transformation --
11.5 The Sylvester Inertia Theorem --
11.6 Test --
11.7 Exercises --
12. Answers to the Tests --
References.
This book covers the material of an introductory course in linear algebra: sets and maps, vector spaces, bases, linear maps, matrices, determinants, systems of linear equations, Euclidean spaces, eigenvalues and eigenvectors, diagonalization of self-adjoint operators, and classification of matrices. The book is written for beginners. Its didactic features (the "book within a book" and multiple choice tests with commented answers) make it especially suitable for self-study
81-8128-187-X
Mathematics
Algebras, Linear
Algebra
Linear algebra - New Delhi Springer India Ltd., 2004 - IX+204 p. 23 cm ;Pbk
58 illustrations
1. Sets and Maps --
1.1 Sets --
1.2 Maps --
1.3 Test --
1.4 Remarks on the Literature --
1.5 Exercises --
2. Vector Spaces --
2.1 Real Vector Spaces --
2.2 Complex Numbers and Complex Vector Spaces --
2.3 Vector Subspaces --
2.4 Test --
2.5 Fields --
2.6 What Are Vectors? --
2.7 Complex Numbers 400 Years Ago --
2.8 Remarks on the Literature --
2.9 Exercises --
3. Dimension --
3.1 Linear Independence --
3.2 The Concept of Dimension --
3.3 Test --
3.4 Proof of the Basis Extension Theorem and the Exchange Lemma --
3.5 The Vector Product --
3.6 The 'Steinitz Exchange Theorem' --
3.7 Exercises --
4. Linear Maps --
4.1 Linear Maps --
4.2 Matrices --
4.3 Test --
4.4 Quotient Spaces --
4.5 Rotations and Reflections in the Plane --
4.6 Historical Aside --
4.7 Exercises --
5. Matrix Calculus --
5.1 Multiplication --
5.2 The Rank of a Matrix --
5.3 Elementary Transformations --
5.4 Test --
5.5 How Does One Invert a Matrix? --
5.6 Rotations and Reflections (continued) --
5.7 Historical Aside --
5.8 Exercises --
6. Determinants --
6.1 Determinants --
6.2 Determination of Determinants --
6.3 The Determinant of the Transposed Matrix --
6.4 Determinantal Formula for the Inverse Matrix --
6.5 Determinants and Matrix Products --
6.6 Test --
6.7 Determinant of an Endomorphism --
6.8 The Leibniz Formula --
6.9 Historical Aside --
6.10 Exercises --
7. Systems of Linear Equations --
7.1 Systems of Linear Equations --
7.2 Cramer's Rule --
7.3 Gaussian Elimination --
7.4 Test --
7.5 More on Systems of Linear Equations --
7.6 Captured on Camera! --
7.7 Historical Aside --
7.8 Remarks on the Literature --
7.9 Exercises --
8. Euclidean Vector Spaces --
8.1 Inner Products --
8.2 Orthogonal Vectors --
8.3 Orthogonal Maps --
8.4 Groups --
8.5 Test --
8.6 Remarks on the Literature --
8.7 Exercises --
9. Eigenvalues --
9.1 Eigenvalues and Eigenvectors --
9.2 The Characteristic Polynomial --
9.3 Test --
9.4 Polynomials --
9.5 Exercises --
10. The Principal Axes Transformation --
10.1 Self-Adjoint Endomorphisms --
10.2 Symmetric Matrices --
10.3 The Principal Axes Transformation for Self-Adjoint Endomorphisms --
10.4 Test --
10.5 Exercises --
11. Classification of Matrices --
11.1 What Is Meant by 'Classification'? --
11.2 The Rank Theorem --
11.3 The Jordan Normal Form --
11.4 More on the Principal Axes Transformation --
11.5 The Sylvester Inertia Theorem --
11.6 Test --
11.7 Exercises --
12. Answers to the Tests --
References.
This book covers the material of an introductory course in linear algebra: sets and maps, vector spaces, bases, linear maps, matrices, determinants, systems of linear equations, Euclidean spaces, eigenvalues and eigenvectors, diagonalization of self-adjoint operators, and classification of matrices. The book is written for beginners. Its didactic features (the "book within a book" and multiple choice tests with commented answers) make it especially suitable for self-study
81-8128-187-X
Mathematics
Algebras, Linear
Algebra