Appendix C Theory and procedures
1.1 Sets and functions
Theorem 1.1.23 Invertible is equivalent to bijective
2.1 Systems of linear equations
Theorem 2.1.12 Row equivalence theorem
2.2 Gaussian elimination
Theorem 2.2.10 Row equivalent matrix forms
2.3 Solving linear systems
Theorem 2.3.5 Solving linear systems
Corollary 2.3.6 \(0\text{,}\) \(1\text{,}\) or \(\infty\)-many solutions
Corollary 2.3.10 Solutions to homogeneous equations
3.1 Matrices and their arithmetic
Theorem 3.1.19 Column method of matrix multiplication
Theorem 3.1.21 Row method of matrix multiplication
3.2 Algebra of matrices
Theorem 3.2.1 Properties of matrix addition, multiplication and scalar multiplication
Theorem 3.2.4 Additive identities, additive inverses, and multiplicative identities
Corollary 3.2.5 Additive cancellation of matrices
Theorem 3.2.8 Matrix algebra abnormalities
Corollary 3.2.9 Failure of multiplicative cancellation
Theorem 3.2.11 Properties of matrix transposition
3.3 Invertible matrices
Theorem 3.3.2 Inverses are unique
Theorem 3.3.3 Solving with invertible matrices
Theorem 3.3.5 Inverses of \(2\times 2\) matrices
Theorem 3.3.6 Invertibility of products
Theorem 3.3.13 Properties of matrix powers
Theorem 3.3.14 Inverse and transpose
3.4 The invertibility theorem
Theorem 3.4.2 Elementary matrix formulas
Theorem 3.4.3 Inverses of elementary matrices
Theorem 3.4.5 Invertibility theorem
Theorem 3.4.8 Invertibility of triangular matrices
Theorem 3.4.9 Inverse algorithm
Theorem 3.4.10 Product of elementary matrices algorithm
Corollary 3.4.11 Left-inverse if and only if right-inverse
Corollary 3.4.12 Invertibility of product equivalence
Corollary 3.4.13 Row equivalence and invertible matrices
Corollary 3.4.15 Uniqueness of reduced row echelon form
3.5 The determinant
Theorem 3.5.5 Determinant of triangular matrices
Corollary 3.5.6 Determinant of identity matrices
Theorem 3.5.8 Expansion along rows
Theorem 3.5.9 Determinant and transposition
Corollary 3.5.10 Expansion along columns
Theorem 3.5.13 Zero rows/columns, swapping rows/columns, identical rows/columns
Theorem 3.5.16 Adjoint formula
Theorem 3.5.19 Row operations and determinant
Corollary 3.5.22 Determinant and products of elementary matrices
Theorem 3.5.24 Determinant and invertibility
Theorem 3.5.25 Determinant is multiplicative
Theorem 3.5.26 Invertibility theorem (extended cut)
4.1 Real vector spaces
Theorem 4.1.13 Basic vector space properties
4.2 Linear transformations
Theorem 4.2.5 Basic properties of linear transformations
Theorem 4.2.9 Matrix transformations I
Theorem 4.2.13 Rotation is a linear transformation
Theorem 4.2.17 Reflection is a linear transformation
Theorem 4.2.22 Composition of linear transformations
4.3 Subspaces
Theorem 4.3.6 Intersection of subspaces
Theorem 4.3.14 Null space and image
Theorem 4.3.15 Nullspace and injectivity
4.4 Span and linear independence
Theorem 4.4.4 Spans are subspaces
Procedure 4.4.10 Investigating linear independence
Procedure 4.4.13 Investigating linear independence of functions
4.5 Bases and dimension
Theorem 4.5.7 Basis equivalence
Theorem 4.5.11 Basis bounds
Theorem 4.5.15 Contracting and expanding to bases
Corollary 4.5.16 Street smarts
Corollary 4.5.17 Dimension of subspaces
Theorem 4.5.18 Dimension theory compendium
4.6 Rank-nullity theorem and fundamental spaces
Theorem 4.6.2 Rank-nullity
Theorem 4.6.6 Fundamental spaces
Procedure 4.6.7 Computing bases of fundamental spaces
Theorem 4.6.8 Invertibility theorem (supersized)
Procedure 4.6.9 Contracting and extending to bases of \(\R^n\)
4.7 Isomorphisms
Theorem 4.7.1 Bases and linear transformations
Corollary 4.7.4 Matrix transformations II
Theorem 4.7.9 Properties preserved by isomorphisms
5.1 Inner product spaces
Theorem 5.1.3 Weighted dot product
Theorem 5.1.8 Dot product method of matrix multiplication
Theorem 5.1.9 Evaluation inner products on \(P_n\)
Theorem 5.1.11 Integral inner product
Theorem 5.1.16 Basic properties of norm and distance
Theorem 5.1.17 Cauchy-Schwarz inequality
Theorem 5.1.18 Triangle Inequalities
5.2 Orthogonal bases and orthogonal projection
Theorem 5.2.2 Orthogonal implies linearly independent
Theorem 5.2.4 Existence of orthonormal bases
Procedure 5.2.5 Gram-Schmidt procedure
Theorem 5.2.6 Calculating with orthogonal bases
Theorem 5.2.7 Orthogonal matrices
Theorem 5.2.10 Orthogonal complement
Theorem 5.2.12 Orthogonal projection theorem
6.1 Coordinate vectors and isomorphisms
Theorem 6.1.7 Coordinate vectors for orthogonal bases
Theorem 6.1.9 Coordinate vector transformation
Procedure 6.1.10 Contracting and extending to bases in general spaces
Theorem 6.1.12 Isomorphism compendium
6.2 Matrix representations of linear transformations
6.3 Change of basis
Theorem 6.3.2 Change of basis for coordinate vectors
Theorem 6.3.3 Change of basis matrix properties
Theorem 6.3.4 Change of basis for transformations
6.4 Eigenvectors and eigenvalues
Theorem 6.4.3 Characteristic polynomial theorem
Procedure 6.4.5 Computing eigenspaces of a matrix
6.5 Diagonalization
Theorem 6.5.2 Diagonalizability theorem
Procedure 6.5.3 Deciding whether a linear transformation is diagonalizable
Theorem 6.5.5 Algebraic and geometric multiplicity theorem
Theorem 6.5.11 Properties of conjugation
Theorem 6.5.15 Properties of similarity
Theorem 6.5.17 Invertibility theorem (final version)