Matrices and Linear Transformations"Comprehensive . . . an excellent introduction to the subject." — Electronic Engineer's Design Magazine. This introductory textbook, aimed at sophomore- and junior-level undergraduates in mathematics, engineering, and the physical sciences, offers a smooth, in-depth treatment of linear algebra and matrix theory. The major objects of study are matrices over an arbitrary field. Contents include Matrices and Linear Systems; Vector Spaces; Determinants; Linear Transformations; Similarity: Part I and Part II; Polynomials and Polynomial Matrices; Matrix Analysis; and Numerical Methods. The first seven chapters, which require only a first course in calculus and analytic geometry, deal with matrices and linear systems, vector spaces, determinants, linear transformations, similarity, polynomials, and polynomial matrices. Chapters 8 and 9, parts of which require the student to have completed the normal course sequence in calculus and differential equations, provide introductions to matrix analysis and numerical linear algebra, respectively. Among the key features are coverage of spectral decomposition, the Jordan canonical form, the solution of the matrix equation AX = XB, and over 375 problems, many with answers. |
Contents
Matrices and Linear Systems | 1 |
22 | 51 |
Vector Spaces | 67 |
Determinants | 104 |
Linear Transformations | 124 |
Part I | 175 |
Polynomials and Polynomial Matrices | 215 |
Part II | 236 |
Matrix Analysis | 255 |
Numerical Methods | 272 |
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Common terms and phrases
a₁ a₁x a₂ AE Fx algebraic B₁ b₂ C₂ characteristic polynomial characteristic value compute Consider converges corollary defined Definition diagonal matrix dim CS(A divisor E₁ echelon form elementary matrices elements equation Example Exercise exists F[x]nxn field F form a basis function hence Hermitian Hermitian matrix integers inverse Jordan blocks Jordan canonical form k₁ linear operator linear transformation linearly independent lower triangular m x n matrix m₁(x minimum polynomial monic n x n N₁ nilpotent nonsingular matrix NS(A one-to-one orthogonal matrix orthonormal basis P₁ p₁(x P₂ partitioned proof of Theorem properties Prove Theorem R₁ R₂ rank reader real numbers result row equivalent row-reduced echelon RS(A satisfies Section sequence set of vectors Show Smith canonical solution space over F Span subset subspace symmetric system AX t-invariant triangular matrix unique unitary V₁ vector space X₁ zero α₁ λ₁