Appendix D Theory and procedures
0.2 Functions
Theorem 0.2.11 Invertible is equivalent to bijective
0.6 Complex numbers
Theorem 0.6.6 Basic properties of complex arithmetic
Theorem 0.6.10 Properties of conjugation and modulus
0.7 Polynomials
Theorem 0.7.3 Basic properties of polynomials
Corollary 0.7.4 Polynomial equality via coefficients
Theorem 0.7.6 Basic properties of degree
Theorem 0.7.7 Fundamental theorem of algebra
1.1 Vector space structure of \(\R^n\)
Procedure 1.1.6 Verifying vector space axioms
Theorem 1.1.9 Basic vector space properties
1.2 Inner product structure of \(\R^n\)
Theorem 1.2.6 Weighted dot product
2.1 Systems of linear equations
Theorem 2.1.16 Row equivalence theorem
2.2 Gaussian elimination
Procedure 2.2.11 Gaussian elimination
Procedure 2.2.13 Gauss-Jordan elimination
Theorem 2.2.15 Row equivalent matrix forms
2.3 Solving linear systems
Procedure 2.3.6 Solving linear systems
Corollary 2.3.13 Solutions to homogeneous equations
3.1 Matrix arithmetic
Theorem 3.1.16 Vector space structure of \(M_{mn}\)
Theorem 3.1.25 Dot product and matrix multiplication
Theorem 3.1.27 Column method of matrix multiplication
Theorem 3.1.29 Row method of matrix multiplication
3.2 Matrix algebra
Theorem 3.2.1 Properties of matrix multiplication
Theorem 3.2.5 Multiplicative identities
Theorem 3.2.6 Additive cancellation of matrices
Theorem 3.2.9 Matrix algebra abnormalities
Corollary 3.2.10 Failure of multiplicative cancellation
Theorem 3.2.12 Properties of matrix transposition
3.3 Invertible matrices
Theorem 3.3.2 Inverses are unique
Theorem 3.3.3 Cancellation with invertible matrices
Corollary 3.3.4 Solving with invertible matrices
Theorem 3.3.6 Inverses of \(2\times 2\) matrices
Theorem 3.3.7 Products of invertible matrices
Corollary 3.3.8 Products of invertible matrices
Theorem 3.3.14 Properties of matrix powers
Theorem 3.3.15 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 3.4.9 Invertibility of triangular matrices
Procedure 3.4.10 Inverse algorithm
Procedure 3.4.11 Product of elementary matrices algorithm
Corollary 3.4.15 Left-inverse if and only if right-inverse
Corollary 3.4.16 Invertibility of product equivalence
Corollary 3.4.17 Row equivalence and invertible matrices
Corollary 3.4.19 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 3.5.14 Zero rows/columns, swapping rows/columns, identical rows/columns
Theorem 3.5.17 Adjoint matrix formula
Theorem 3.5.20 Row operations and determinant
Corollary 3.5.23 Determinant and products of elementary matrices
Theorem 3.5.25 Determinant and invertibility
Theorem 3.5.26 Determinant is multiplicative
Theorem 3.5.27 Invertibility theorem
4.1 Subspaces
Procedure 4.1.4 Two-step proof for subspaces
Theorem 4.1.7 Subspaces are vector spaces
Theorem 4.1.8 Intersection of subspaces
Theorem 4.1.11 Null spaces of matrices
Theorem 4.1.15 Null space and linear systems
Corollary 4.1.16 Null space and linear systems
Theorem 4.1.22 Matrix subspaces
4.2 Span and linear independence
Theorem 4.2.6 Spans are subspaces
Theorem 4.2.12 Linear independence of finite sets
Procedure 4.2.15 Linear independence of a finite set
Theorem 4.2.17 Invertibility theorem
4.3 Bases
Theorem 4.3.4 Basis equivalence
Procedure 4.3.5 One-step technique for bases
Procedure 4.3.11 By-inspection basis technique
4.4 Dimension
Theorem 4.4.10 Spanning set bound
Theorem 4.4.11 Basis bounds
Corollary 4.4.12 Infinite dimensional spaces
Procedure 4.4.14 Computing dimension
Theorem 4.4.15 Contracting and expanding to bases
Corollary 4.4.16 Street smarts
Corollary 4.4.17 Dimension of subspaces
Theorem 4.4.20 Dimension theory compendium
4.5 Fundamental spaces
Theorem 4.5.5 Column space and linear systems
Theorem 4.5.7 Fundamental spaces and row operations
Procedure 4.5.10 Fundamental spaces
Corollary 4.5.13 Rank-nullity for matrices
Procedure 4.5.18 Contracting and extending to bases of \(\R^n\)
Theorem 4.5.21 Invertibility theorem
5.1 Linear transformations
Theorem 5.1.9 Elementary linear transformation proofsZero, identity, and scaling transformations
Theorem 5.1.10 Basic properties of linear transformations
Procedure 5.1.12 One-step technique for transformations
Theorem 5.1.14 Linear matrix operations
Theorem 5.1.16 Conjugation is linear
Theorem 5.1.21 Bases and linear transformations
Theorem 5.1.26 Matrix transformations
Procedure 5.1.29 Standard matrix
Theorem 5.1.34 Rotation is a linear transformation
Theorem 5.1.38 Reflection is a linear transformation
Theorem 5.1.41 Composition of linear transformations
5.2 Null space, image, and isomophisms
Theorem 5.2.3 Null space and image
Theorem 5.2.6 Null space and image of matrix transformation
Theorem 5.2.9 Rank-nullity
Theorem 5.2.12 Injectivity and surjectivity
Corollary 5.2.14 Dimension, injectivity, surjectivity
Theorem 5.2.16 Inverse of isomorphism is linear
Theorem 5.2.18 Isomorphism equivalence
Theorem 5.2.20 Properties preserved by isomorphisms
Corollary 5.2.23 Isomorphic to \(\R^n\)
Theorem 5.2.24 Invertibility theorem
5.3 Coordinate vectors
Procedure 5.3.4 Computing coordinate vectors
Theorem 5.3.11 Coordinate vector transformation
Procedure 5.3.12 Contracting and extending to bases in general spaces
5.4 Matrix representations of linear transformations
Theorem 5.4.3 Standard matrix as a matrix representation
Theorem 5.4.6 Defining property of matrix representation
Theorem 5.4.9 Computing with matrix representations
5.5 Change of basis
Theorem 5.5.2 Change of basis for coordinate vectors
Theorem 5.5.5 Change of basis matrix properties
Procedure 5.5.12 Change of basis computational tips
Theorem 5.5.13 Change of basis for transformations
Procedure 5.5.16 Computing the standard matrix using change of basis
Theorem 5.5.23 Similarity and matrix representations
6.1 Eigenvectors and eigenvalues
Theorem 6.1.13 Eigenvectors of a linear transformation
Theorem 6.1.14 Eigenvectors of matrices
Corollary 6.1.16 Eigenvalues of a matrix
Procedure 6.1.19 Computing eigenspaces of a matrix
Corollary 6.1.24 Null space as eigenspace
Procedure 6.1.25 Computing eigenspaces of a linear transformation
Theorem 6.1.27 Characteristic polynomial
Corollary 6.1.29 Number of eigenvalues
Theorem 6.1.30 Invertibility theorem
6.2 Diagonalization
Theorem 6.2.2 Diagonalizabilty: basis of eigenvectors
Theorem 6.2.7 Linear independence of eigenvectors
Corollary 6.2.8 Diagonalizable if distinct eigenvalues
Corollary 6.2.11 Independence of eigenspaces
Theorem 6.2.12 Diagonalizability: dimension of eigenspaces
Procedure 6.2.13 Deciding whether a linear transformation is diagonalizable
Procedure 6.2.17 Deciding whether a matrix is diagonalizable
Corollary 6.2.19 Diagonalizabilty and similarity
Theorem 6.2.20 Properties of similarity
Theorem 6.2.22 Properties of conjugation
Theorem 6.2.29 Algebraic and geometric multiplicity
Corollary 6.2.30 Algebraic and geometric multiplicity
7.1 Inner product spaces
Theorem 7.1.8 Basic properties of norm and distance
Theorem 7.1.9 Cauchy-Schwarz inequality
Theorem 7.1.10 Triangle Inequalities
7.2 Orthogonal bases
Theorem 7.2.2 Orthogonal implies linearly independent
Procedure 7.2.6 Gram-Schmidt procedure
Corollary 7.2.7 Existence of orthonormal bases
Theorem 7.2.8 Calculating with orthogonal bases
Theorem 7.2.11 Coordinate vectors for orthogonal bases
Theorem 7.2.16 Orthogonal matrices
Theorem 7.2.18 Orthonormal change of basis
7.3 Orthogonal projection
Theorem 7.3.5 Orthogonal complement
Theorem 7.3.7 Fundamental spaces and orthogonal complements
Theorem 7.3.9 Orthogonal projection theorem
Theorem 7.3.12 Orthogonal projection onto a line
Theorem 7.3.16 Orthogonal projection onto a plane
Corollary 7.3.22 Complement of a complement
Corollary 7.3.23 Orthgonal projection is linear
Theorem 7.3.31 Least-squares matrix formula
7.4 The spectral theorem
Theorem 7.4.2 Self-adjoint operators and symmetry
Corollary 7.4.3 Self-adjoint operators and symmetry
Theorem 7.4.4 Eigenvalues of self-adjoint operators
Corollary 7.4.5 Eigenvalues of self-adjoint operators
Theorem 7.4.8 Spectral theorem for self-adjoint operators
Corollary 7.4.11 Spectral theorem for symmetric matrices
Procedure 7.4.12 Orthogonal diagonalization
