You are here

Linear Algebra

Sterling K. Berberian
Publisher: 
Dover Publications
Publication Date: 
2014
Number of Pages: 
358
Format: 
Paperback
Price: 
24.95
ISBN: 
9780486780559
Category: 
Textbook
[Reviewed by
Allen Stenger
, on
10/10/2014
]

This book is two textbooks in one. The first half is a straightforward (although very theoretical) introduction to linear algebra and matrices at the lower-division undergraduate level, while the second half is a much more in-depth look at the theory of linear algebra at the upper-division undergraduate level. The two courses are not intended to be presented back-to-back, but separated by a course in linear abstract algebra. The present volume is a 2014 Dover reprint of the 1992 Oxford edition, with two new pages of errata and notes.

The first half, although theoretical, is not abstract. Most work is done in R2 and R3, and the book never ventures beyond finite-dimensional spaces. It uses some of the language of abstract algebra, including quotient spaces and a few commutative diagrams. It is proof-oriented, with few worked examples, and the exercises are mostly proofs with a little drill included. The presentation starts in terms of linear mappings, and only after these are established does it begin using matrix language. The second half is a collection of special topics in the deeper theory of linear algebra. This again is oriented toward proofs.

The linear algebra market has shifted since this book was first published, and it’s not clear what course today could use the first half of the book. It omits all applications, and it omits such now-common topics as Gaussian elimination, special factorizations such as LU and QR, Singular Value Decomposition, and any numerical considerations. The second half is still useful, and contains hard-to-find items such as a thorough development of the theory of determinants and a development of rational and Jordan canonical forms that does not require much algebra background.


Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.

  • PART I
    • 1 Vector spaces
      • 1.1 Motivation (vectors in 3-space)
      • 1.2 Rn and Cn
      • 1.3 Vector spaces: the axioms, some examples
      • 1.4 Vector spaces: first consequences of the axioms
      • 1.5 Linear combinations of vectors
      • 1.6 Linear subspaces
    • 2 Linear mappings
      • 2.1 Linear mappings
      • 2.2 Linear mappings and linear subspaces: kernel and range
      • 2.3 Spaces of linear mappings: L(V, W) and L(V)
      • 2.4 Isomorphic vector spaces
      • 2.5 Equivalence relations and quotient sets
      • 2.6 Quotient vector spaces
      • 2.7 The first isomorphism theorem
    • 3 Structure of vector spaces
      • 3.1 Linear subspace generated by a subset
      • 3.2 Linear dependence
      • 3.3 Linear independence
      • 3.4 Finitely generated vector spaces
      • 3.5 Basis, dimension
      • 3.6 Rank + nullity = dimension
      • 3.7 Applications of R + N = D
      • 3.8 Dimension of L(V, W)
      • 3.9 Duality in vector spaces
    • 4 Matrices
      • 4.1 Matrices
      • 4.2 Matrices of linear mappings
      • 4.3 Matrix multiplication
      • 4.4 Algebra of matrices
      • 4.5 A model for linear mappings
      • 4.6 Transpose of a matrix
      • 4.7 Calculating the rank
      • 4.8 When is a linear system solvable?
      • 4.9 An example
      • 4.10 Change of basis, similar matrices
    • 5 Inner product spaces
      • 5.1 Inner product spaces, Euclidean spaces
      • 5.2 Duality in inner product spaces
      • 5.3 The adjoint of a linear mapping
      • 5.4 Orthogonal mappings and matrices
    • 6 Determinants (2 x 2 and 3 x 3)
      • 6.1 Determinant of a 2 x 2 matrix
      • 6.2 Cross product of vectors in R3
      • 6.3 Determinant of a 3 x 3 matrix
      • 6.4 Characteristic polynomial of a matrix (2 x 2 or 3 x 3)
      • 6.5 Diagonalizing 2 x 2 symmetric real matrices
      • 6.6 Diagonalizing 3 x 3 symmetric real matrices
      • 6.7 A geometric application (conic sections)
  • PART II
    • 7 Determinants (n x n)
      • 7.1 Alternate multilinear forms
      • 7.2 Determinant of a linear mapping
      • 7.3 Determinant of a square matrix
      • 7.4 Cofactors
    • 8 Similarity (Act I)
      • 8.1 Similarity
      • 8.2 Eigenvalues and eigenvectors
      • 8.3 Characteristic polynomial
    • 9 Euclidean spaces (Spectral Theory)
      • 9.1 Invariant and reducing subspaces
      • 9.1 Bounds of a linear mapping
      • 9.3 Bounds of a self-adjoint mapping, Spectral Theorem
      • 9.4 Normal linear mappings in Euclidean spaces
    • 10 Equivalence of matrices over a PIR
      • 10.1 Unimodular matrices
      • 10.2 Preview of the theory of equivalence
      • 10.3 Equivalence: existence of a diagonal form
      • 10.4 Equivalence: uniqueness of the diagonal form
    • 11 Similarity (Act II)
      • 11.1 Invariant factors, Fundamental theorem of similarity
      • 11.2 Companion matrix, Rational canonical form
      • 11.3 Hamilton-Cayley theorem, minimal polynomial
      • 11.4 Elementary divisors, Jordan canonical form
      • 11.5 Appendix: proof that Mn(F)[t] = Mn(F[t])
    • 12 Unitary spaces
      • 12.1 Complex inner product spaces, unitary spaces
      • 12.2 Orthogonality
      • 12.3 Orthonormal bases, isomorphism
      • 12.4 Adjoint of a linear mapping
      • 12.5 Invariant and reducing subspaces
      • 12.6 Special linear mappings and matrices
      • 12.7 Normal linear mappings, Spectral Theorem
      • 12.8 The Spectral Theorem: another way
    • 13 Tensor products
      • 13.1 Tensor product VW of vector spaces
      • 13.2 Tensor product ST of linear mappings
      • 13.3 Matrices of tensor products
  • Appendix A: Foundations
    • A.1 A dab of logic
    • A.2 Set notations
    • A.3 Functions
    • A.4 The axioms for a field
  • Appendix B: Integral domains, factorization theory
    • B.1 The field of fractions of an integral domain
    • B.2 Divisibility in an integral domain
    • B.3 Principal ideal rings
    • 8.4 Euclidean integral domains
    • 4.5 Factorization in overfields
  • Appendix C: Weierstrass-Bolzano theorem
  • Index of notations
  • Index
  • Errata and Comments

Dummy View - NOT TO BE DELETED