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 Systems of linear equations
Example 1.1.2 Linear and nonlinear equations
Example 1.1.6 Solutions to elementary systems
Example 1.1.7 Lines and planes review
Example 1.1.9 Solutions to elementary systems (again)
Example 1.1.14 Complete example
1.2 Gaussian elimination
Example 1.2.4 Row echelon versus reduced row echelon form
Example 1.2.9 Row echelon form
Example 1.2.11 Reduced row echelon form
Example 1.2.13 Row echelon form is not unique
1.3 Solving linear systems
Example 1.3.2 Free and leading variables
Example 1.3.3 Solving linear systems
2.1 Matrix arithmetic
Example 2.1.6 Matrix equality
Example 2.1.14 Matrix linear combinations
Example 2.1.15 Expressing matrix as a linear combination
Example 2.1.20 Matrix multiplication
Example 2.1.23 Matrix multiplication via dot product
Example 2.1.27 Column and row methods
Example 2.1.31 Transpose
2.3 Invertible matrices
Example 2.3.5 Invertible matrices
Example 2.3.13 Matrix polynomials
2.4 The invertibility theorem
Example 2.4.4 Inverses of elementary matrices
Example 2.4.8 Triangular \(3\times 3\) matrices
Example 2.4.12 Inverse algorithms
2.5 The determinant
Example 2.5.24 Determinant via row reduction
3.1 Real vector spaces
Example 3.1.3 Vector space of \(m\times n\) matrices
Example 3.1.4 Vector space of real \(n\)-tuples
Example 3.1.9 Zero vector space
Example 3.1.10 The vector space of infinite real sequences
Example 3.1.11 Real-valued functions
Example 3.1.13 Vector space of positive real numbers
Example 3.1.15 Vector linear combination
3.2 Linear transformations
Example 3.2.4 Model of linear transformation proof
Example 3.2.15 Rotation matrices
Example 3.2.18 Visualizing reflection and rotation
Example 3.2.22 Orthogonal projection onto line
Example 3.2.26 Visualizing orthogonal projection
Example 3.2.28 Transposition is linear
Example 3.2.29 Left-shift transformation
3.3 Subspaces
Example 3.3.14 Lines and planes
Example 3.3.17 Trace-zero symmetric, and skew-symmetric \(2\times 2\) matrices
3.4 Null space and image
Example 3.4.4 Matrix transformation
Example 3.4.7 The derivative (calculus refresher)
Example 3.4.12 Image computation
Example 3.4.13 A differential equation
Example 3.4.17 Lines and planes (again)
3.5 Span and linear independence
Example 3.5.3 Examples in \(\R^2\)
Example 3.5.8 Spanning sets are not unique
Example 3.5.11 Elementary examples
Example 3.5.13 Linear independence
Example 3.5.16 Linear independence in \(P\)
Example 3.5.19 Linear independence of functions
3.6 Bases
Example 3.6.3 Some nonstandard bases
Example 3.6.4 \(P\) has no finite basis
Example 3.6.6 Basis for \(\R_{>0}\)
Example 3.6.9 One-step technique for \(\R^3\)
Example 3.6.10 One-step technique for \(P_1\)
Example 3.6.14 By-inspection basis technique
Example 3.6.17 Composition of reflections
Example 3.6.20 Standard matrix computation
Example 3.6.21 Rotation matrices revisited
3.7 Dimension
Example 3.7.7 Familiar vector spaces
Example 3.7.8 Subspace of \(P_3\)
Example 3.7.9 Dimension of symmetric matrices
Example 3.7.14 Dimension of subspace
Example 3.7.15 Subspaces of \(\R^3\)
3.8 Rank-nullity theorem and fundamental spaces
Example 3.8.3 Rank-nullity: verification
Example 3.8.4 Rank-nullity: application
4.1 Coordinate vectors
Example 4.1.5 Standard bases
Example 4.1.6 Reorderings of standard bases
Example 4.1.7 Nonstandard bases
4.2 Matrix representations of linear transformations
Example 4.2.12 Two representations of orthogonal projection
4.3 Change of basis
Example 4.3.17 Orthogonal projection (again)
4.4 Eigenvectors and eigenvalues
Example 4.4.8 Zero and identity transformations
Example 4.4.9 Reflection
Example 4.4.10 Rotation
Example 4.4.11 Transposition
Example 4.4.12 Differentiation
Example 4.4.17 Rotation (again)
Example 4.4.24 Transposition (again)
4.5 Diagonalization
Example 4.5.16 Transposition
Example 4.5.24 Diagonalizable: matrix powers
Example 4.5.25 Diagonalizable: matrix polynomials
5.1 Inner product spaces
Example 5.1.5 Dot product on \(\R^4\)
Example 5.1.6 Weighted dot product
Example 5.1.7 Why the weights must be positive
Example 5.1.11 Evaluation at \(-1, 0, 1\)
Example 5.1.13 Integral inner product
Example 5.1.15 Norm with respect to dot product
Example 5.1.16 Norm with respect to weighted dot product
Example 5.1.17 Norm with respect to integral inner product
Example 5.1.19 Unit vectors
Example 5.1.30 Weighted dot product distance
Example 5.1.31 Evaluation inner product distance
Example 5.1.32 Integral inner product and distance
5.2 Orthogonal bases
Example 5.2.12 Orthogonal bases
Example 5.2.19 Orthonormal change of basis: \(\R^n\)
Example 5.2.20 Orthonormal change of basis: polynomials
5.3 Orthogonal projection
Example 5.3.15 Projection onto a line \(\ell\subseteq \R^3\)
Example 5.3.17 Projection onto planes in \(\R^3\)
Example 5.3.19 Best fitting line
5.4 The spectral theorem
Example 5.4.6 Symmetric \(2\times 2\) matrices
Example 5.4.7 Symmetric \(4\times 4\) matrix
Example 5.4.13 Orthogonal diagonalization
Example 5.4.15 Orthogonal projections are self-adjoint
