Appendix D Examples
1.1 Sets and functions
Example 1.1.15 Arithmetic operations as functions
Example 1.1.20 Role of domain and codomain in injectivity and surjectivity
1.2 Logic
Example 1.2.7 Modeling “Every positive number has a square-root”
Example 1.2.10 The limit does not exist
1.3 Proof techniques
Example 1.3.3 Proof: invertible is equivalent to bijective
Example 1.3.4 Proof by contradiction
Example 1.3.8 Weak induction
Example 1.3.10 Strong induction
2.1 Systems of linear equations
Example 2.1.2 Linear and nonlinear equations
Example 2.1.6 Solutions to elementary systems
2.2 Gaussian elimination
Example 2.2.4 Row echelon versus reduced row echelon form
Example 2.2.11 Row echelon form is not unique
2.3 Solving linear systems
3.1 Matrices and their arithmetic
Example 3.1.22 Matrix multiplication
3.3 Invertible matrices
Example 3.3.12 Matrix polynomials
3.4 The invertibility theorem
3.5 The determinant
Example 3.5.23 Determinant via row reduction
4.1 Real vector spaces
4.2 Linear transformations
Example 4.2.15 Rotation matrices
Example 4.2.18 Transposition is linear
Example 4.2.19 Left-shift transformation
4.3 Subspaces
Example 4.3.20 Vector spaces of symmetric/skew-symmetric matrices
Example 4.3.21 Image computation
Example 4.3.23 A differential equation
4.4 Span and linear independence
Example 4.4.3 Examples in \(\R^2\)
Example 4.4.7 Spanning sets are not unique
Example 4.4.12 Linear independence
Example 4.4.15 Linear independence of functions
4.5 Bases and dimension
Example 4.5.3 Some nonstandard bases
Example 4.5.4 \(P\) has no finite basis
Example 4.5.6 Basis for \(\R_{>0}\)
Example 4.5.8 One-step technique for bases
4.6 Rank-nullity theorem and fundamental spaces
Example 4.6.3 Rank-nullity: verification
Example 4.6.4 Rank-nullity: application
4.7 Isomorphisms
Example 4.7.3 Composition of reflections
Example 4.7.5 Rotation matrices revisited
5.1 Inner product spaces
Example 5.1.4 Dot product on \(\R^4\)
Example 5.1.5 Weighted dot product
Example 5.1.6 Why the weights must be positive
Example 5.1.10 Evaluation at \(-1, 0, 1\)
Example 5.1.12 Integral inner product
6.1 Coordinate vectors and isomorphisms
Example 6.1.4 Standard bases
Example 6.1.5 Reorderings of standard bases
Example 6.1.6 Nonstandard bases
Example 6.1.8 Orthogonal bases
6.5 Diagonalization