Linearity
We began this book by learning how to systematically solve systems of linear algebraic equations. This “elementary” problem formed our launching pad for developing the fundamentals of linear algebra. In its initial form, matrices and vectors were the prim
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Linearity We began this book by learning how to systematically solve systems of linear algebraic equations. This “elementary” problem formed our launching pad for developing the fundamentals of linear algebra. In its initial form, matrices and vectors were the primary focus of our study, but the theory was developed in a sufficiently general and abstract form that it can be immediately used in many other useful situations — particularly infinitedimensional function spaces. Indeed, applied mathematics deals, not just with algebraic equations, but also with differential equations, difference equations, integral equations, stochastic systems, differential delay equations, control systems, and many other types — only a few of which, unfortunately, can be adequately developed in this introductory text. It is now time to assemble what we have learned about linear algebraic systems and place the results in a suitably general framework that will lead to insight into the key principles that govern all linear systems arising in mathematics and its applications. The most basic underlying object of linear systems theory is the vector space, and we have already seen that the elements of vector spaces can be vectors, or functions, or even vector-valued functions. The seminal ideas of span, linear independence, basis, and dimension are equally applicable and equally vital in more general contexts, particularly function spaces. Just as vectors in Euclidean space are prototypes for elements of general vector spaces, matrices are also prototypes for more general objects, known as linear functions. Linear functions are also known as linear maps or, when one is dealing with function spaces, linear operators, and include linear differential operators, linear integral operators, function evaluation, and many other basic operations. Linear operators on infinite-dimensional function spaces are the basic objects of quantum mechanics. Each quantum mechanical observable (mass, energy, momentum) is formulated as a linear operator on an infinitedimensional Hilbert space — the space of wave functions or states of the system, [54]. It is remarkable that quantum mechanics is an entirely linear theory, whereas classical and relativistic mechanics are inherently nonlinear. The holy grail of modern physics — the unification of general relativity and quantum mechanics — is to resolve the apparent incompatibility of the microscopic linear and macroscopic nonlinear physical regimes. In geometry, linear functions are interpreted as linear transformations of space (or spacetime), and, as such, lie at the foundations of motion of bodies, such as satellites and planets; computer graphics and games; video, animation, and movies; and the mathematical formulation of symmetry. Many familiar geometrical transformations, including rotations, scalings and stretches, reflections, projections, shears, and screw motions, are linear. But including translational motions requires a slight extension of linearity, known as an affine transformation. The basic geometry of linear and affi
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