8: Appendices
- Page ID
- 228844
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\dsum}{\displaystyle\sum\limits} \)
\( \newcommand{\dint}{\displaystyle\int\limits} \)
\( \newcommand{\dlim}{\displaystyle\lim\limits} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\(\newcommand{\longvect}{\overrightarrow}\)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In Appendix A, complex arithmetic is developed far enough to find nth roots. In Appendix B, methods of proof are discussed, while Appendix C presents mathematical induction. Finally, Appendix D describes the properties of polynomials in elementary terms.
- 8.A: Complex Numbers
- This page explores complex numbers, beginning with their necessity for solving polynomial equations and introducing the fundamental theorem of algebra. It defines complex numbers and operations, absolute value, and geometric interpretations on the complex plane. The polar form and Euler's formula are discussed, along with De Moivre's theorem for calculating powers and roots.
- 8.B: Proofs
- This page covers essential logical methods used in various disciplines, such as science and mathematics. It details direct proof, case analysis, proof by contradiction, and the use of counterexamples to disprove statements, including specific examples like the primality of Mersenne primes and disproving inequalities. The Pigeonhole Principle and irrationality of \(e\) are also discussed.
- 8.C: Mathematical Induction
- This page covers the principle of mathematical induction, illustrating its application through proofs of various mathematical statements. It shows how to verify propositions for natural numbers by establishing truth for a base case and proving implications for succeeding cases. Key examples include the sum of odd numbers, polynomial functions, geometrical series, inequalities, and congruences.
- 8.D: Polynomials
- This page covers polynomials, defining their coefficients, variables, and key concepts like leading coefficients and polynomial degree. It introduces the Remainder Theorem, explaining polynomial division, roots, and the Factor Theorem, which links factors to roots. The page illustrates polynomial factoring, counting roots and multiplicities, and notes that a polynomial of degree \(n\) can have at most \(n\) roots.
- 8.E: LU-Factorization
- This page covers the LU factorization of matrices, where a matrix \(A\) can be expressed as \(A = LU\) with \(L\) (lower triangular) and \(U\) (upper triangular). It describes using Gaussian elimination for obtaining \(LU\) and the role of permutation matrices to handle row interchanges for effective factorization. The uniqueness of \(L\) and \(U\) is confirmed when \(A\) has full rank, and techniques for transforming matrices into row-echelon form are detailed.
- 8.F: Proof of the Cofactor Expansion Theorem
- This page provides an inductive definition of the determinant for \(n \times n\) matrices, starting with \(1 \times 1\) matrices, and explains cofactor expansion. It includes methods for computing determinants, illustrates properties under row operations, and outlines key theorems regarding the effects of row interchanges and identical rows on the determinant.
- 8.G: Isomorphisms and Composition
- This page discusses the concept of isomorphisms in vector spaces, explaining that different representations can reflect the same underlying space. It covers the criteria for two finite-dimensional spaces to be isomorphic via linear transformations and bases. The significance of linear transformations, their compositions, and the existence of unique inverses under isomorphisms are emphasized.
- 8.H: A Theorem about Differential Equations
- This page covers the significance of differential equations in science, introducing linear differential equations with constant coefficients and their solution sets as vector spaces. It explores complex-valued functions and establishes relationships between dimensions of solution spaces. The properties of linear operators on vector spaces are detailed, including key functions and differentiation rules.
- 8.I: More on Linear Recurrences
- This page covers linear recurrences through the lens of vector spaces and linear transformations, establishing the representation of sequences as vectors. It discusses the linearity, injectiveness, and surjectiveness of transformation \(T\) on these sequences, while introducing the Vandermonde matrix and the shift operator.
- 8.J: Orthogonality
- This page covers orthogonality in \(\mathbb{R}^n\), detailing the properties of orthogonal vector sets and their independence, which simplifies vector expansions. It aims to extend the concept of bases to orthogonal bases, highlighting their benefits. Additionally, it discusses orthogonal complements, diagonalization, methods for computing eigenvalues, singular value decomposition, and various practical applications across different fields.
- 1.1: Prelude to Orthogonality
- 1.2: Orthogonal Diagonalization
- 1.2E: Orthogonal Diagonalization Exercises
- 1.3: Positive Definite Matrices
- 1.3E: Positive Definite Matrices Exercises
- 1.4: QR-Factorization
- 1.4E: QR-Factorization Exercises
- 1.5: Computing Eigenvalues
- 1.5E: Computing Eigenvalues Exercises
- 1.6: The Singular Value Decomposition
- 1.6E: The Singular Value Decomposition Exercises
- 1.7: Complex Matrices
- 1.7E: Complex Matrices Exercises
- 1.8: An Application to Linear Codes over Finite Fields
- 1.8E: An Application to Linear Codes over Finite Fields Exercises
- 1.9: An Application to Quadratic Forms
- 1.9E: An Application to Quadratic Forms Exercises
- 1.10: An Application to Constrained Optimization
- 1.11: An Application to Statistical Principal Component Analysis
- 8.K: Change of Basis
- This page explains how linear transformations correlate with matrices, defining the matrix transformation \(T_A\) for an \(m \times n\) matrix \(A\). It highlights that any linear transformation from \(\mathbb{R}^n\) to \(\mathbb{R}^m\) can be represented as a matrix transformation, which is uniquely determined by the transformation on the standard basis.
- 8.L: Canonical Forms
- This page covers matrix transformations through row and column operations, leading to different canonical forms. It highlights the process of achieving reduced row-echelon form via row operations and Smith canonical form through column operations. The concept of similarity transformations for square matrices is introduced, leading to Jordan canonical forms, with a focus on finding the Jordan form using an appropriate basis.
- 8.M: Inner Product Spaces
- This page introduces essential linear algebra concepts like inner products, norms, and orthogonal sets of vectors, emphasizing their geometric interpretation and the implications for orthogonal diagonalization. It explores isometries and their significance in preserving distances during transformations, with applications in Fourier approximation. To enhance comprehension, the section includes exercises related to these themes.
- 1.1: Prelude to Inner Product Spaces
- 1.2: Inner Products and Norms
- 1.2E: Inner Products and Norms Exercises
- 1.3: Orthogonal Sets of Vectors
- 1.3E: Orthogonal Sets of Vectors Exercises
- 1.4: Orthogonal Diagonalization
- 1.4E: Orthogonal Diagonalization Exercises
- 1.5: Isometries
- 1.5E: Isometries Exercises
- 1.6: An Application to Fourier Approximation
- 1.6E: An Application to Fourier Approximation Exercises