Linear Algebra for the Young Mathematician

Cover
Title page
Preface
Part I . Vector spaces
	Chapter 1. The basics
		1.1. The vector space Fⁿ
		1.2. Linear combinations
		1.3. Matrices and the equation ??=?
		1.4. The basic counting theorem
		1.5. Matrices and linear transformations
		1.6. Exercises
	Chapter 2. Systems of linear equations
		2.1. The geometry of linear systems
		2.2. Solving systems of equations—setting up
		2.3. Solving linear systems—echelon forms
		2.4. Solving systems of equations—the reduction process
		2.5. Drawing some consequences
		2.6. Exercises
	Chapter 3. Vector spaces
		3.1. The notion of a vector space
		3.2. Linear combinations
		3.3. Bases and dimension
		3.4. Subspaces
		3.5. Affine subspaces and quotient vector spaces
		3.6. Exercises
	Chapter 4. Linear transformations
		4.1. Linear transformations I
		4.2. Matrix algebra
		4.3. Linear transformations II
		4.4. Matrix inversion
		4.5. Looking back at calculus
		4.6. Exercises
	Chapter 5. More on vector spaces and linear transformations
		5.1. Subspaces and linear transformations
		5.2. Dimension counting and applications
		5.3. Bases and coordinates: vectors
		5.4. Bases and matrices: linear transformations
		5.5. The dual of a vector space
		5.6. The dual of a linear transformation
		5.7. Exercises
	Chapter 6. The determinant
		6.1. Volume functions
		6.2. Existence, uniqueness, and properties of the determinant
		6.3. A formula for the determinant
		6.4. Practical evaluation of determinants
		6.5. The classical adjoint and Cramer’s rule
		6.6. Jacobians
		6.7. Exercises
	Chapter 7. The structure of a linear transformation
		7.1. Eigenvalues, eigenvectors, and generalized eigenvectors
		7.2. Polynomials in cT
		7.3. Application to differential equations
		7.4. Diagonalizable linear transformations
		7.5. Structural results
		7.6. Exercises
	Chapter 8. Jordan canonical form
		8.1. Chains, Jordan blocks, and the (labelled) eigenstructure picture of cT
		8.2. Proof that cT has a Jordan canonical form
		8.3. An algorithm for Jordan canonical form and a Jordan basis
		8.4. Application to systems of first-order differential equations
		8.5. Further results
		8.6. Exercises
Part II . Vector spaces with additional structure
	Chapter 9. Forms on vector spaces
		9.1. Forms in general
		9.2. Usual types of forms
		9.3. Classifying forms I
		9.4. Classifying forms II
		9.5. The adjoint of a linear transformation
		9.6. Applications to algebra and calculus
		9.7. Exercises
	Chapter 10. Inner product spaces
		10.1. Definition, examples, and basic properties
		10.2. Subspaces, complements, and bases
		10.3. Two applications: symmetric and Hermitian forms, and the singular value decomposition
		10.4. Adjoints, normal linear transformations, and the spectral theorem
		10.5. Exercises
	Appendix A. Fields
		A.1. The notion of a field
		A.2. Fields as vector spaces
	Appendix B. Polynomials
		B.1. Statement of results
		B.2. Proof of results
	Appendix C. Normed vector spaces and questions of analysis
		C.1. Spaces of sequences
		C.2. Spaces of functions
	Appendix D. A guide to further reading
	Index
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