0. Introductory.- 0.1 Physical Dimensions.- 0.2 Mathematical Dimensions.- 0.3 Overview.- Exercises.- 1. The Mathematical Foundations of Science and Engineering.- 1.1 The Inadequacy of Real Numbers.- 1.1.1 The Error of Substitution.- 1.1.2 The Problem with Linear Spaces.- 1.1.3 Nondimensionalization.- 1.1.3 Dimensioned Algebras.- 1.2 The Mathematics of Dimensioned Quantities.- 1.2.1 Axiomatic Development.- 1.2.2 Constructive Approach.- 1.2.3 Constraints on Exponentiation.- 1.2.4 The Dimensional Basis.- 1.2.5 Dimensional Logarithms.- 1.2.6 The Basis-Independence Principle.- 1.2.7 Symmetries of Dimensioned Quantities.- 1.2.8 Images.- 1.3 Conclusions.- Exercises.- 2. Dimensioned Linear Algebra.- 2.1 Vector Spaces and Linear Transformations.- 2.2 Terminology and Dimensional Inversion.- 2.3 Dimensioned Scalars.- 2.4 Dimensioned Vectors.- 2.5 Dimensioned Matrices.- Exercises.- 3. The Theory of Dimensioned Matrices.- 3.1 The Dimensional Freedom of Multipliable Matrices.- 3.2 Endomorphic Matrices and the Matrix Exponential.- 3.3 Square Matrices, Inverses, and the Determinant.- 3.4 Squarable Matrices and Eigenstructure.- 3.5 Dimensionally Symmetric Multipliable Matrices.- 3.6 Dimensionally Hankel and Toeplitz Matrices.- 3.7 Uniform, Half Uniform, and Dimensionless Matrices.- 3.8 Conclusions.- 3.A Appendix: The n + m ? 1 Theorem.- Exercises.- 4. Norms, Adjoints, and Singular Value Decomposition.- 4.1 Norms for Dimensioned Spaces.- 4.1.1 Wand Norms.- 4.1.2 Extrinsic Norms.- 4.2 Dimensioned Singular Value Decomposition (DSVD).- 4.3 Adjoints.- 4.4 Norms for Nonuniform Matrices.- 4.5 A Control Application.- 4.6 Factorization of Symmetric Matrices.- Exercises.- 5. Aspects of the Theory of Systems.- 5.1 Differential and Difference Equations.- 5.2 State-Space Forms.- 5.3 Canonical Forms.- 5.4 Transfer Functions and Impulse Responses.- 5.5 Duals and Adjoints.- 5.6 Stability.- 5.7 Controllability, Observability, and Grammians.- 5.8 Expectations and Probability Densities.- Exercises.- 6. Multidimensional Computational Methods.- 6.1 Computers and Engineering.- 6.1.1 A Software Environment for Dimensioned Linear Algebra.- 6.1.2 Overview.- 6.2 Representing and Manipulating Dimensioned Scalars.- 6.2.1 The Numeric and Dimensional Components of a Scalar.- 6.2.2 The Dimensional Basis.- 6.2.3 Numerical Representations and Uniqueness.- 6.2.4 Scalar Operations.- 6.2.5 Input String Conversion.- 6.2.6 Output and Units Conversion.- 6.2.7 Binary Relations.- 6.2.8 Summary of Scalar Methods.- 6.3 Dimensioned Vectors.- 6.3.1 Dimensioned Vectors and Dimension Vectors.- 6.3.2 Representing Dimensioned Vectors.- 6.3.3 Vector Operations.- 6.3.4 Summary of Vectors.- 6.4 Representing Dimensioned Matrices.- 6.4.1 Arrays versus Matrices.- 6.4.2 The Domain/Range Matrix Representation.- 6.1.3 Allowing Geometric and Matrix Algebra Interpretations.- 6.4.4 Input Conversion.- 6.4.5 Output Conversion.- 6.4.6 Special Classes of Dimensioned Matrices.- 6.4.7 Identity and Zero Matrices.- 6.4.8 Scalar and Vector Conversion to Matrices.- 6.4.9 Summary of the Matrix Representation.- 6.5 Operations on Dimensioned Matrices.- 6.5.1 Matrix Addition, Subtraction, Similarity, and Equality.- 6.5.2 Block Matrices.- 6.5.3 Matrix Multiplication.- 6.5.4 Gaussian Elimination.- 6 5.5 The Determinant and Singularity.- 6.5.6 The Trace.- 6.5.7 Matrix Inverse.- 6.5.8 Matrix Transpose.- 6.5.9 Eigenstructure Decomposition.- 6.5.10 Singular Value Decomposition.- 6.6 Conclusions.- Exercises.- 7. Forms of Multidimensional Relationships.- 7.1 Goals.- 7.2 Operations.- 7.3 Procedure.- Exercises.- 8. Concluding Remarks.- 9. Solutions to Odd-Numbered Exercises.- References.
