Appendix E Examples
0.1 Sets
0.2 Functions
Example 0.2.3 Arithmetic operations as functions
Example 0.2.8 Role of domain and codomain in injectivity and surjectivity
0.4 Logic
Example 0.4.7 Modeling “Every positive number has a square-root”
Example 0.4.10 The limit does not exist
0.5 Proof techniques
Example 0.5.3 Proof: invertible is equivalent to bijective
Example 0.5.4 Proof by contradiction
Example 0.5.8 Weak induction
Example 0.5.10 Strong induction
0.6 Complex numbers
1.1 Vector space structure of \(\R^n\)
Example 1.1.11 Vector space of infinite sequences
Example 1.1.12 Vector space of positive reals
Example 1.1.14 Linear combination
Example 1.1.15 Linear combination
1.2 Inner product structure of \(\R^n\)
Example 1.2.2 Dot product on \(\R^4\)
Example 1.2.5 Weighted dot product
Example 1.2.7 Why the weights must be positive
2.1 Systems of linear equations
Example 2.1.2 Linear and nonlinear equations
Example 2.1.5 Visualizing lines in \(\R^2\)
Example 2.1.7 Visualizing planes in \(\R^3\)
Example 2.1.11 Solutions to elementary systems
Example 2.1.12 Solutions to elementary systems (again)
Example 2.1.17 Complete example
2.2 Gaussian elimination
Example 2.2.7 Row echelon versus reduced row echelon form
Example 2.2.12 Row echelon form
Example 2.2.14 Reduced row echelon form
Example 2.2.16 Row echelon form is not unique
2.3 Solving linear systems
Example 2.3.2 Free and leading variables
Example 2.3.3 Solving linear systems
Example 2.3.7 Video: solving linear systems
Example 2.3.9 Video: solving linear systems
Example 2.3.14 Vector parametrization
3.1 Matrix arithmetic
Example 3.1.7 Matrix equality
Example 3.1.18 Matrix linear combinations
Example 3.1.19 Expressing matrix as a linear combination
Example 3.1.24 Matrix multiplication
Example 3.1.26 Matrix multiplication via dot product
Example 3.1.30 Column and row methods
Example 3.1.31 Video example of matrix multiplication
Example 3.1.35 Transpose
3.2 Matrix algebra
Example 3.2.2 Video example: proving matrix equalities
Example 3.2.13 Video example: proving matrix equalities
3.3 Invertible matrices
Example 3.3.5 Invertible matrices
Example 3.3.13 Matrix polynomials
3.4 The invertibility theorem
Example 3.4.4 Inverses of elementary matrices
Example 3.4.8 Triangular \(3\times 3\) matrices
Example 3.4.12 Inverse algorithms
Example 3.4.13 Video example: inverse algorithm
3.5 The determinant
Example 3.5.24 Determinant via row reduction
4.1 Subspaces
Example 4.1.5 Video example: deciding if \(W\subseteq V\) is a subspace
Example 4.1.13 Subspace as null space
Example 4.1.20 Trace-zero symmetric, and skew-symmetric \(2\times 2\) matrices
4.2 Span and linear independence
Example 4.2.3 Examples in \(\R^2\)
Example 4.2.4 Video example: computing span
Example 4.2.9 Spanning sets are not unique
Example 4.2.13 Elementary examples
Example 4.2.16 Linear independence
4.3 Bases
Example 4.3.3 Some nonstandard bases
Example 4.3.6 One-step technique for \(\R^3\)
Example 4.3.7 Video example: deciding if a set is a basis (\(\R^n\))
Example 4.3.9 Video example: deciding if a set is a basis (\(M_{nn}\))
Example 4.3.12 By-inspection basis technique
Example 4.3.13 By-inspection basis technique
4.4 Dimension
Example 4.4.5 Dimensions of familiar spaces
Example 4.4.6 Dimension of subspace
Example 4.4.7 Dimension of symmetric matrices
Example 4.4.8 Video example: computing dimension
Example 4.4.13 Infinite dimensional space
Example 4.4.18 Dimension of subspace
Example 4.4.19 Subspaces of \(\R^3\)
4.5 Fundamental spaces
Example 4.5.3 Fundamental spaces: elementary example
Example 4.5.9 Column space and row reduction
Example 4.5.11 Fundamental spaces
Example 4.5.14 Rank-nullity
Example 4.5.15 Using the rank-nullity theorem
Example 4.5.16 Video example: fundamental spaces
Example 4.5.19 Video example: contracting to a basis
5.1 Linear transformations
Example 5.1.5 Nonlinear function
Example 5.1.13 Linear transformation: one-step technique
Example 5.1.17 Left-shift transformation on \(\R^\infty\)
Example 5.1.18 Video examples: deciding if \(T\) is linear
Example 5.1.24 Bases and transformations
Example 5.1.30 Standard matrix computation
Example 5.1.36 Rotation matrices
Example 5.1.39 Visualizing reflection and rotation
Example 5.1.42 Composition of reflections
5.2 Null space, image, and isomophisms
Example 5.2.7 Matrix transformation
Example 5.2.10 Rank-nullity application
Example 5.2.11 Rank-nullity application
Example 5.2.19 Transposition is an isomorphism
5.3 Coordinate vectors
Example 5.3.5 Standard bases
Example 5.3.6 Reorderings of standard bases
Example 5.3.7 Nonstandard bases
Example 5.3.8 Video example: coordinate vectors
5.4 Matrix representations of linear transformations
Example 5.4.2 Matrix representation
Example 5.4.5 Different choice of bases
Example 5.4.10 Computing with matrix representations
Example 5.4.11 Video example: matrix representations
5.5 Change of basis
Example 5.5.3 Change of basis
Example 5.5.4 Change of basis
Example 5.5.7 Change of basis, \(B\) standard
Example 5.5.9 Change of basis, \(B\) standard
Example 5.5.10 Video example: change of basis matrix
Example 5.5.15 Matrix representations and change of basis
Example 5.5.17 Reflection in \(\R^2\)
Example 5.5.18 Video example: change of basis for transformations
Example 5.5.20 Video example: change of basis and reflection
6.1 Eigenvectors and eigenvalues
Example 6.1.8 Zero and identity transformations
Example 6.1.9 Scaling transformations
Example 6.1.10 Reflection
Example 6.1.11 Rotation
Example 6.1.12 Transposition
Example 6.1.17 Rotation (again)
Example 6.1.20 Compute eigenspaces
Example 6.1.21 Compute eigenspaces
Example 6.1.23 Compute eigenspaces
Example 6.1.26 Transposition (again)
6.2 Diagonalization
Example 6.2.15 Transposition
Example 6.2.23 Diagonalizable: matrix powers
Example 6.2.24 Diagonalizable: matrix polynomials
7.1 Inner product spaces
Example 7.1.2 Norm with respect to dot product
Example 7.1.3 Norm with respect to weighted dot product
Example 7.1.5 Unit vectors
Example 7.1.15 Weighted dot product distance
7.2 Orthogonal bases
Example 7.2.12 Orthogonal bases
Example 7.2.19 Orthonormal change of basis: \(\R^n\)
Example 7.2.20 Orthonormal change of basis: polynomials
7.3 Orthogonal projection
Example 7.3.13 Orthogonal projection onto line
Example 7.3.17 Visualizing orthogonal projection
Example 7.3.20 Video example: orthogonal projection in function space
Example 7.3.24 Projection onto a line \(\ell\subseteq \R^3\)
Example 7.3.26 Projection onto planes in \(\R^3\)
Example 7.3.28 Best fitting line
7.4 The spectral theorem
Example 7.4.6 Symmetric \(2\times 2\) matrices
Example 7.4.7 Symmetric \(4\times 4\) matrix
Example 7.4.13 Orthogonal diagonalization
Example 7.4.15 Orthogonal projections are self-adjoint
