Linear Algebra and Its Application
- Page ID
- 190540
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 0: Why Linear Algebra?
- Linear algebra is an elegant topic focusing on vectors, matrices, and linear transformations, which are essential across diverse fields such as mathematics, engineering, computer science, and economics. It outlines key topics like solving linear equations, properties of vector spaces, linear independence, and eigenvalues, while emphasizing real-world applications.
- 1: Solving Systems of Linear Equations
- This chapter discusses systems of linear equations, covering definitions, classifications, and geometric interpretations in two and three variables. It introduces matrix notation and row reduction methods for solving these systems, including row echelon forms and variable implications. Additionally, it addresses homogeneous systems, matrix rank, and their significance.
- 2: Matrices
- This chapter covers fundamental matrix operations such as addition, scalar multiplication, and particularly matrix multiplication, which is emphasized as vital. It details how to find the ijth entry of a product, discusses properties of multiplication, introduces the identity matrix and inverses, and explains elementary matrices and LU factorization. Exercises are included to reinforce learning.
- 3: Determinants
- This chapter explains the importance of determinants in square matrices, covering their properties, applications, and geometric interpretations. It discusses their relation to matrix inverses and Cramer's Rule, emphasizes the effects of row operations on determinants, and includes examples for clarity. Furthermore, it highlights the geometric interpretation of determinants as volumes, enhancing comprehension of their defining properties and relevance in multivariable calculus.
- 4: Vector Spaces - Rⁿ
- This chapter details vector concepts, covering operations like dot and cross products, lines and planes in \(\mathbb{R}^3\), and spanning sets. It discusses linear independence, matrix spaces (row, column, null), orthogonal vectors/matrices, the Gram-Schmidt process for orthonormal sets, orthogonal projections, and least squares approximation. The page includes exercises for practice at the end of each section.
- 4.1: Vector Spaces
- 4.2: Subspaces
- 4.3: Review of Vectors
- 4.4: Dot and Cross Product
- 4.5: Lines and Planes
- 4.6: Spanning Sets in Rⁿ
- 4.7: Linear Independence
- 4.8: Subspaces
- 4.9: Subspaces and Bases
- 4.10: Row, Column and Null Spaces
- 4.11: Dot Products and Orthogonality
- 4.12: Orthogonal Vectors and Matrices
- 4.13: Gram-Schmidt Process
- 4.14: Orthogonal Complements
- 4.15: Orthogonal Projections
- 4.16: Orthogonal Projection
- 4.17: Least Squares Approximation
- 5: Linear Transformations
- This chapter explores the link between linear transformations and matrices, covering topics like matrix transformations, one-to-one and onto transformations, and their identification. It discusses matrix multiplication, including composition and scalar multiplication, and introduces matrix inverses for equation solving. Overall, it highlights the utility of linear algebra in analyzing transformations via matrices.
- 6: Eigenvalues and Eigenvectors
- This chapter explains eigenvalues and eigenvectors, providing methods for their computation, their significance in diagonalization, and applications in dynamical systems. It discusses Markov chains, particularly in the context of Google's PageRank algorithm, and serves as an educational resource on fundamental matrix theories and practical uses.
- 7: Honors Projects
- This page discusses stochastic matrices and their applications in difference equations and Markov chains, notably in Google's PageRank algorithm. Key concepts include eigenvalues, the Perron-Frobenius theorem, and the least squares method for inconsistent matrices. It also explores applications in linear recurrences, differential equations, computer graphics, and economic models.
- 7.1: Stochastic Matrices
- 7.2: An Application to Markov Chains
- 7.3: The Method of Least Squares
- 7.4: An Application to Linear Recurrences
- 7.5: An Application to Systems of Differential Equations
- 7.6: An Application to Computer Graphics
- 7.7: An Application to Input-Output Economic Models
- 7.8: The Complex Number System