Applied Linear Algebra: The Decoupling Principle
A useful reference, this book could easily be subtitled: All the Linear Algebra I Learned from Doing Physics that I Wished Somebody had Taught Me First. Built upon the principles of diagonalization and superposition, it contains many important physical applications—such as population growth, normal modes of oscillations, waves, Markov chains, stability analysis, signal processing, and electrostatics—in order to demonstrate the incredible power of linear algebra in the world. The underlying ideas of breaking a vector into modes, and of decoupling a complicated system by suitable choice of linear coordinates, are emphasized throughout the book. Chapter topics most useful to professional engineers and physicists include—but are not limited to—the wave equation, continuos spectra, fourier transforms, and Green's function. For electrical engineers, physicists, and mechanical engineers.
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The Decoupling Principle
Vector Spaces and Bases
Linear Transformations and Operators
9 other sections not shown
algebra arbitrary axis called Chapter column complex vector space compute conjugate consider converges corresponding decay Decoupling define Definition denoted derivative diagonal diagonalizable differential equation dimension dimensional spaces Dirichlet boundary conditions eigen eigenfunctions eigenvalues eigenvalues and eigenvectors eigenvectors element entries equal example Exercises exponential Figure Find the Green's finite fixed point formula Fourier coefficients Fourier series Fourier transform func Green's function half line Hermitian operator identity infinite dimensional initial condition inner product space integral kernel internal direct sum Ker(L limit linear combination linear operator linear transformation linearly independent Neumann boundary conditions nonzero orthogonal orthonormal basis periodic function polynomial power vector problem Proof real numbers real symmetric real vector space result rotation scalar multiplication Section Show solution solve span square-integrable standard basis standard inner product subspace Suppose Theorem tion traveling waves unstable modes values variable wave equation whole line zero