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Unformatted text preview: Vector Spaces The idea of vectors dates back to the middle 1800’s, but our current understanding of the concept waited until Peano’s work in 1888. Even then it took many years to understand the importance and generality of the ideas involved. This one underlying idea can be used to describe the forces and accelerations in Newtonian mechanics and the potential functions of electromagnetism and the states of systems in quantum mechanics and the least-square fitting of experimental data and much more. 6.1 The Underlying Idea What is a vector? If your answer is along the lines “something with magnitude and direction” then you have some- thing to unlearn. Maybe you heard this definition in a class that I taught. If so, I lied; sorry about that. At the very least I didn’t tell the whole truth. Does an automobile have magnitude and direction? Does that make it a vector? The idea of a vector is far more general than the picture of a line with an arrowhead attached to its end. That special case is an important one, but it doesn’t tell the whole story, and the whole story is one that unites many areas of mathematics. The short answer to the question of the first paragraph is A vector is an element of a vector space. Roughly speaking, a vector space is some set of things for which the operation of addition is defined and the operation of multiplication by a scalar is defined. You don’t necessarily have to be able to multiply two vectors by each other or even to be able to define the length of a vector, though those are very useful operations and will show up in most of the interesting cases. You can add two cubic polynomials together: ( 2- 3 x + 4 x 2- 7 x 3 ) + (- 8- 2 x + 11 x 2 + 9 x 3 ) makes sense, resulting in a cubic polynomial. You can multiply such a polynomial by* 17 and it’s still a cubic polynomial. The set of all cubic polynomials in x forms a vector space and the vectors are the individual cubic polynomials. The common example of directed line segments (arrows) in two or three dimensions fits this idea, because you can add such arrows by the parallelogram law and you can multiply them by numbers, changing their length (and reversing direction for negative numbers). Another, equally important example consists of all ordinary real-valued functions of a real variable: two such functions can be added to form a third one, and you can multiply a function by a number to get another function. The example of cubic polynomials above is then a special case of this one. A complete definition of a vector space requires pinning down these ideas and making them less vague. In the end, the way to do that is to express the definition as a set of axioms. From these axioms the general properties of vectors will follow....
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