Appendix EL Examples
ΒΆSection WILA What is Linear Algebra?
Example 1 Trail Mix Packaging
Section SSLE Solving Systems of Linear Equations
Example 1 Solving two (nonlinear) equations
Example 2 Notation for a system of equations
Example 3 Three typical systems
Example 4 Three equations, one solution
Example 5 Three equations, infinitely many solutions
Section RREF Reduced Row-Echelon Form
Example 1 A matrix
Example 2 Notation for systems of linear equations
Example 3 Augmented matrix for Archetype A
Example 4 Two row-equivalent matrices
Example 5 Three equations, one solution, reprised
Example 6 A matrix in reduced row-echelon form
Example 7 A matrix not in reduced row-echelon form
Example 8 Solutions for Archetype B
Example 9 Solutions for Archetype A
Example 10 Solutions for Archetype E
Section TSS Types of Solution Sets
Example 1 Reduced row-echelon form notation
Example 2 Describing infinite solution sets, Archetype I
Example 3 Free and dependent variables
Example 4 Counting free variables
Example 5 One solution gives many, Archetype D
Section HSE Homogeneous Systems of Equations
Example 1 Archetype C as a homogeneous system
Example 2 Homogeneous, unique solution, Archetype B
Example 3 Homogeneous, infinite solutions, Archetype A
Example 4 Homogeneous, infinite solutions, Archetype D
Example 5 Null space elements of Archetype I
Example 6 Computing a null space, #1
Example 7 Computing a null space, #2
Section NM Nonsingular Matrices
Example 1 A singular matrix, Archetype A
Example 2 A nonsingular matrix, Archetype B
Example 3 An identity matrix
Example 4 Singular matrix, row-reduced
Example 5 Nonsingular matrix, row-reduced
Example 6 Null space of a singular matrix
Example 7 Null space of a nonsingular matrix
Section VO Vector Operations
Example 1 Vector equality for a system of equations
Example 2 Addition of two vectors in \(\complex{4}\)
Example 3 Scalar multiplication in \(\complex{5}\)
Section LC Linear Combinations
Example 1 Two linear combinations in \(\complex{6}\)
Example 2 Archetype B as a linear combination
Example 3 Archetype A as a linear combination
Example 4 Vector form of solutions for Archetype D
Example 5 Vector form of solutions
Example 6 Vector form of solutions for Archetype I
Example 7 Vector form of solutions for Archetype L
Example 8 Particular solutions, homogeneous solutions, Archetype D
Section SS Spanning Sets
Example 1 A basic span
Example 2 Span of the columns of Archetype A
Example 3 Span of the columns of Archetype B
Example 4 Spanning set of a null space
Example 5 Null space directly as a span
Example 6 Span of the columns of Archetype D
Section LI Linear Independence
Example 1 Linearly dependent set in \(\complex{5}\)
Example 2 Linearly independent set in \(\complex{5}\)
Example 3 Linearly independent, homogeneous system
Example 4 Linearly dependent, homogeneous system
Example 5 Linearly dependent, \(r\) and \(n\)
Example 6 Large linearly dependent set in \(\complex{4}\)
Example 7 Linearly dependent columns in Archetype A
Example 8 Linearly independent columns in Archetype B
Example 9 Linear independence of null space basis
Example 10 Null space spanned by linearly independent set, Archetype L
Section LDS Linear Dependence and Spans
Example 1 Reducing a span in \(\complex{5}\)
Example 2 Casting out vectors
Example 3 Reducing a span in \(\complex{4}\)
Example 4 Reworking elements of a span
Section O Orthogonality
Example 1 Computing some inner products
Example 2 Computing the norm of some vectors
Example 3 Two orthogonal vectors
Example 4 Standard Unit Vectors are an Orthogonal Set
Example 5 An orthogonal set
Example 6 Gram-Schmidt of three vectors
Example 7 Orthonormal set, three vectors
Example 8 Orthonormal set, four vectors
Section MO Matrix Operations
Example 1 Addition of two matrices in \(M_{23}\)
Example 2 Scalar multiplication in \(M_{32}\)
Example 3 Transpose of a \(3\times 4\) matrix
Example 4 A symmetric \(5\times 5\) matrix
Example 5 Complex conjugate of a matrix
Section MM Matrix Multiplication
Example 1 A matrix times a vector
Example 2 Matrix notation for systems of linear equations
Example 3 Money's best cities
Example 4 Product of two matrices
Example 5 Matrix multiplication is not commutative
Example 6 Product of two matrices, entry-by-entry
Section MISLE Matrix Inverses and Systems of Linear Equations
Example 1 Solutions to Archetype B with a matrix inverse
Example 2 A matrix without an inverse, Archetype A
Example 3 Matrix inverse
Example 4 Computing a matrix inverse
Example 5 Computing a matrix inverse, Archetype B
Section MINM Matrix Inverses and Nonsingular Matrices
Example 1 Unitary matrix of size 3
Example 2 Unitary permutation matrix
Example 3 Orthonormal set from matrix columns
Section CRS Column and Row Spaces
Example 1 Column space of a matrix and consistent systems
Example 2 Membership in the column space of a matrix
Example 3 Column space, two ways
Example 4 Column space, original columns, Archetype D
Example 5 Column space of Archetype A
Example 6 Column space of Archetype B
Example 7 Row space of Archetype I
Example 8 Row spaces of two row-equivalent matrices
Example 9 Improving a span
Example 10 Column space from row operations, Archetype I
Section FS Four Subsets
Example 1 Left null space
Example 2 Column space as null space
Example 3 Submatrices of extended echelon form
Example 4 Four subsets, no. 1
Example 5 Four subsets, no. 2
Example 6 Four subsets, Archetype G
Section VS Vector Spaces
Example 1 The vector space \(\complex{m}\)
Example 2 The vector space of matrices, \(M_{mn}\)
Example 3 The vector space of polynomials, \(P_n\)
Example 4 The vector space of infinite sequences
Example 5 The vector space of functions
Example 6 The singleton vector space
Example 7 The crazy vector space
Example 8 Properties for the Crazy Vector Space
Section S Subspaces
Example 1 A subspace of \(\complex{3}\)
Example 2 A subspace of \(P_4\)
Example 3 A non-subspace in \(\complex{2}\text{,}\) zero vector
Example 4 A non-subspace in \(\complex{2}\text{,}\) additive closure
Example 5 A non-subspace in \(\complex{2}\text{,}\) scalar multiplication closure
Example 6 Recasting a subspace as a null space
Example 7 A linear combination of matrices
Example 8 Span of a set of polynomials
Example 9 A subspace of \(M_{32}\)
Section LISS Linear Independence and Spanning Sets
Example 1 Linear independence in \(P_4\)
Example 2 Linear independence in \(M_{32}\)
Example 3 Linearly independent set in the crazy vector space
Example 4 Spanning set in \(P_4\)
Example 5 Spanning set in \(M_{22}\)
Example 6 Spanning set in the crazy vector space
Example 7 A vector representation
Section B Bases
Example 1 Bases for \(P_n\)
Example 2 A basis for the vector space of matrices
Example 4 A basis for a subspace of \(M_{22}\)
Example 5 Basis for the crazy vector space
Example 6 Row space basis
Example 7 Reducing a span
Example 8 Columns as Basis, Archetype K
Example 9 Coordinatization relative to an orthonormal basis, \(\complex{4}\)
Example 10 Coordinatization relative to an orthonormal basis, \(\complex{3}\)
Section D Dimension
Example 1 Linearly dependent set in \(P_4\)
Example 2 Dimension of a subspace of \(M_{22}\)
Example 3 Dimension of a subspace of \(P_4\)
Example 4 Dimension of the crazy vector space
Example 5 Vector space of polynomials with unbounded degree
Example 6 Rank and nullity of a matrix
Example 7 Rank and nullity of a square matrix
Section PD Properties of Dimension
Example 1 Bases for \(P_n\text{,}\) reprised
Example 2 Basis by dimension in \(M_{22}\)
Example 3 Sets of vectors in \(P_4\)
Example 4 Rank, rank of transpose, Archetype I
Section DM Determinant of a Matrix
Example 1 Elementary matrices and row operations
Example 2 Some submatrices
Example 3 Determinant of a \(3\times 3\) matrix
Example 4 Two computations, same determinant
Example 5 Determinant of an upper triangular matrix
Section PDM Properties of Determinants of Matrices
Example 1 Determinant by row operations
Example 2 Zero and nonzero determinant, Archetypes A and B
Section EE Eigenvalues and Eigenvectors
Example 1 Some eigenvalues and eigenvectors
Example 2 Polynomial of a matrix
Example 3 Computing an eigenvalue the hard way
Example 4 Characteristic polynomial of a matrix, size 3
Example 5 Eigenvalues of a matrix, size 3
Example 6 Eigenspaces of a matrix, size 3
Example 7 Eigenvalue multiplicities, matrix of size 4
Example 8 Eigenvalues, symmetric matrix of size 4
Example 9 High multiplicity eigenvalues, matrix of size 5
Example 10 Complex eigenvalues, matrix of size 6
Example 11 Distinct eigenvalues, matrix of size 5
Section PEE Properties of Eigenvalues and Eigenvectors
Example 1 Building desired eigenvalues
Section SD Similarity and Diagonalization
Example 1 Similar matrices of size 5
Example 2 Similar matrices of size 3
Example 3 Equal eigenvalues, not similar
Example 4 Diagonalization of Archetype B
Example 5 Diagonalizing a matrix of size 3
Example 6 A non-diagonalizable matrix of size 4
Example 7 Distinct eigenvalues, hence diagonalizable
Example 8 High power of a diagonalizable matrix
Example 9 Fibonacci sequence, closed form
Section LT Linear Transformations
Example 1 A linear transformation
Example 2 Not a linear transformation
Example 3 Linear transformation, polynomials to matrices
Example 4 Linear transformation, polynomials to polynomials
Example 5 Linear transformation from a matrix
Example 6 Matrix from a linear transformation
Example 7 Matrix of a linear transformation
Example 8 Linear transformation defined on a basis
Example 9 Linear transformation defined on a basis
Example 10 Linear transformation defined on a basis
Example 11 Sample pre-images, Archetype S
Example 12 Sum of two linear transformations
Example 13 Scalar multiple of a linear transformation
Example 14 Composition of two linear transformations
Section ILT Injective Linear Transformations
Example 1 Not injective, Archetype Q
Example 2 Injective, Archetype R
Example 3 Injective, Archetype V
Example 4 Nontrivial kernel, Archetype O
Example 5 Trivial kernel, Archetype P
Example 6 Not injective, Archetype Q, revisited
Example 7 Not injective, Archetype O
Example 8 Injective, Archetype P
Example 9 Not injective by dimension, Archetype U
Section SLT Surjective Linear Transformations
Example 1 Not surjective, Archetype Q
Example 2 Surjective, Archetype R
Example 3 Surjective, Archetype V
Example 4 Range, Archetype O
Example 5 Full range, Archetype N
Example 6 Not surjective, Archetype Q, revisited
Example 7 Not surjective, Archetype O
Example 8 Surjective, Archetype N
Example 9 A basis for the range of a linear transformation
Example 10 Not surjective by dimension, Archetype T
Section IVLT Invertible Linear Transformations
Example 1 An invertible linear transformation
Example 2 A non-invertible linear transformation
Example 3 Computing the Inverse of a Linear Transformations
Example 4 Isomorphic vector spaces, Archetype V
Section VR Vector Representations
Example 1 Vector representation in \(\complex{4}\)
Example 2 Vector representations in \(P_2\)
Example 3 Two isomorphic vector spaces
Example 4 Crazy vector space revealed
Example 5 A subspace characterized
Example 6 Multiple isomorphic vector spaces
Example 7 Coordinatizing in \(P_2\)
Example 8 Coordinatization in \(M_{32}\)
Section MR Matrix Representations
Example 1 One linear transformation, three representations
Example 2 A linear transformation as matrix multiplication
Example 3 Matrix product of matrix representations
Example 4 Kernel via matrix representation
Example 5 Range via matrix representation
Example 6 Inverse of a linear transformation via a representation
Section CB Change of Basis
Example 1 Eigenvectors of linear transformation between matrices
Example 2 Eigenvectors of linear transformation between polynomials
Example 3 Change of basis with polynomials
Example 4 Change of basis with column vectors
Example 5 Matrix representations and change-of-basis matrices
Example 6 Matrix representation with basis of eigenvectors
Example 7 Eigenvectors of a linear transformation, twice
Example 8 Complex eigenvectors of a linear transformation
Section OD Orthonormal Diagonalization
Example 1 A normal matrix
Section CNO Complex Number Operations
Example 1 Arithmetic of complex numbers
Example 2 Conjugate of some complex numbers
Example 3 Modulus of some complex numbers
Section SET Sets
Example 1 Set membership
Example 2 Subset
Example 3 Cardinality and Size
Example 4 Set union
Example 5 Set intersection
Example 6 Set complement