7: Applications
- Page ID
- 229794
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 7.1: An Application to Electrical Networks
- This page covers the analysis of electrical networks, emphasizing the use of Ohm's Law (V = RI) and Kirchhoff’s Laws for calculating current. It defines resistors in ohms and voltage sources in volts, explains Kirchhoff's Junction Rule (current conservation at junctions), and the Circuit Rule (sum of voltage changes in a circuit). An example is provided to demonstrate the practical application of these principles for determining unknown currents.
- 7.2: An Application to Input-Output Economic Models
- This page covers Wassily Leontief's Nobel Prize-winning work on mathematical models analyzing economic systems, using an input-output matrix to illustrate industry interactions in a primitive society. It explains the equilibrium condition where expenditures equal revenues and presents a model for equilibrium price structures.
- 7.3: An Application to Markov Chains
- This page covers Markov chains, emphasizing transitions between states determined solely by the current state. It explores transition probabilities, state vectors, and steady-state distributions through various examples, including weather patterns and animal behaviors. By utilizing transition matrices, the text details how to compute future states, showing convergence to long-term probabilities.
- 7.4: An Application to Network Flow
- This page explores the analysis of flow networks, including irrigation systems and traffic networks, emphasizing the Junction Rule where inflows equal outflows. It provides an example of a one-way street network, illustrating the derivation of flow equations at intersections. The text highlights the necessity for positive flow values and discusses practical constraints that may affect these flows.
- 7.5: An Application to Chemical Reactions
- This page explains chemical reactions as the combination of molecules to create new substances, such as the formation of water from hydrogen and oxygen. It emphasizes the importance of balancing reactions, ensuring equal numbers of each type of atom on both sides. The process is illustrated using the example of burning octane, which involves solving equations for the molecule counts and requires positive integer coefficients, thus linking the concept to linear equations.
- 7.6: An Application to Computer Graphics
- This page covers the fundamentals of computer graphics, focusing on image creation through points and instructions for curves, along with matrix transformations to maintain linearity. It provides examples of displaying the letter 'A' and applying transformations like shear and scaling using homogeneous coordinates.
- 7.7: An Application to Linear Recurrences
- This page explores recursive sequences through combinatorial problems, emphasizing linear recurrence relations like \(x_{k+2} = x_{k+1} + x_k\). It introduces methods such as matrix diagonalization to derive explicit formulas for sequences, connecting it to the Fibonacci sequence. The page highlights the Fibonacci sequence's significance in nature and introduces the Binet formula.
- 7.8: An Application to Systems of Differential Equations
- This page covers the concept of differentiable functions and their derivatives, specifically focusing on systems of first-order differential equations. It introduces solving methods through diagonalization, emphasizing matrix representation with eigenvalues and eigenvectors.
- 7.10: An Application to Correlation and Variance
- This page covers key statistical concepts, including sample data analysis, correlation, and sample statistics. It introduces sample mean, variance, and centered data, demonstrating their significance through vector representation and dot products. Correlation is explained using paired data, with caution against assuming causation from observed relationships. The text elaborates on correlation coefficients, their calculation, and the effects of data transformations on statistical measures.
- 7.11: An Application to Differential Equations
- This page covers differentiable functions and differential equations, outlining \(n\)th derivatives and the structure of their solution spaces. It emphasizes that solutions to \(n\)th order equations have dimension \(n\) and discusses characteristic polynomials and basis sets. Additionally, it focuses on second-order equations with complex roots, establishing that functions \(e^{px}\cos(qx)\) and \(e^{px}\sin(qx)\) form a basis for solutions.