\[ \newcommand{R}{\mathbb{R}} \newcommand{defarrow}{\quad \stackrel{\text{def}}{\Longleftrightarrow} \quad} \]

# Linear Methods

This is a developing online course book introducing some basic concepts such as linear vector spaces, metric spaces and Banach and Hilbert spaces. In addition, it is intended to cover some different matrix decomposition methods as a tool for representing the more abstract theory.

**Please note that the lecture notes are under development.** Suggestions and corrections can be sent to mats [dot] ehrnstrom [at] ntnu [dot] no. *No error is too small to correct.*

** A preliminary printout of the lecture notes:** Lecture notes PDF. If you wish to print pages directly from the online version, select the material you wish to print, choose "print selection" (or equivalent), and print (or export to PDF).

## Contents

### Sets, spaces and sequences

#### Sets and operations

- Sets: Basic definition and examples.
- Membership and inclusions: Set element notation, subsets.
- Set operations: Unions and intersections.
- Relations: Cartesian products, relations and functions.
- Invertibility: Surjectivity, injectivity and invertibility of functions.

#### Basic spaces and their topology

- Vector spaces: Defining properties of real and complex vector spaces. Euclidean space and \(C(I,\mathbb R)\).
- Normed spaces: Definition and equivalence of norms. Supremum and different \(l_p\)-norms. The space \(BC(I,\mathbb R)\).
- Metric spaces: Metrics. Normed spaces as metric spaces. The Euclidean and discrete metrics.
- Balls and spheres: Open balls and spheres in metric spaces. Unit balls induced by different norms. The \(l_p\)-spaces.
- Open and closed sets: Interior points, boundary points, open and closed sets.

#### Limits and completions

- Limits: Continuous and sequential limits. Accumulation points. Relationship between limits, the distance function and closures.
- Completeness: Cauchy sequences. Complete metric spaces, Banach spaces, and characterisation of complete subspaces.
- Completions: Isometries, isomorphisms and embeddings. Dense sets. Separability. The completion theorem.

### Linear spaces and transformations

- Subspaces: The notion of a linear subspace.
- Linear dependence: Linear dependence and independence, span and generating sets
- Bases: Hamel bases of a vector space, dimension.
- Schauder Bases: Schauder bases.
- Basis transformations: Change-of-basis matrix. Correspondence with invertible matrices.
- Gaussian elimination: Gaussian elimination and the row echelon form of a matrix. LU and LUP-decompositions. Gauss–Jordan elimination and the reduced row echelon form of a matrix. Determinants.
- Linear transformations: Linear transformations. The vector space \(L(X,Y)\). Characterization of linear transformations on finite-dimensional vector spaces; matrix representations. Kernels and ranks (null-spaces, columns spaces and row spaces). The rank–nullity theorem, reduction on \(\R^n\) (injective means surjective), geometric interpretation and the Fredholm alternative.
- Bounded linear transformations: Boundedness. The operator norm and the normed space \(B(X,Y)\). Functionals, duals, and the Riesz representation theorem. \(B(X,Y)\) Banach for \(Y\) Banach. Equivalence of boundedness and continuity for linear operators. Finite-dimensional linear mappings are bounded. Kernels of continuous mappings are closed.

### Solving differential equations

- General existence theorems: Initial-value problems. Formulation as first-order systems. Peano's theorem. Lipschitz continuity. Contractions and the Banach fixed-point theorem; the Picard–Lindelöf theorem and Picard iteration.
- Spectral theory: Constant-coefficient linear ODEs: eigenvalues, eigenvectors, spectrum; characterization of solution spaces; the exponential map and solution formulas. Cayley–Hamilton. Generalized eigenspaces, spectral decompositions and the Jordan normal form. Note on the spectral theorem for hermitian matrices.

### Hilbert space theory

- Inner-product spaces: Inner-product spaces and their properties. The Cauchy–Schwarz inequality, parallelogram law and polarization identity. Hilbert spaces. Convex sets and the closest point property.
- Orthogonality: Orthogonality. The projection and Riesz representation theorems. Orthonormal systems, Bessel's inequality and the Fourier series theorem.
- Adjoints and decompositions: Adjoints and self-adjoint (Hermitian, symmetric) operators. Unitary operators and orthogonal matrices. The spectral theorem. Positive definiteness and decompositions of positive definite matrices. The singular value decomposition and the pseudoinverse. Gram–Schmidt orthogonalization and QR-decomposition.