Topologie und Analysis: Eine Einfuhrung in die by B. Booss

By B. Booss

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Sample text

The plane determined by the vectors 2a and b has twice the weight (or double the area measure) of the plane determined by the vectors a and b. As with vectors, this weight is for now only a relative measure within planes of the same attitude. ) In linear algebra, the orientation and the area measure are both well represented by the determinant of a matrix made of the two spanning vectors a and b of the plane: the orientation is its sign, the area measure its weight (both relative to orientation and area measure of the basis used to specify the coordinates of a and b).

Yet already we will have much to add to its usual structure. By the end of this chapter you will realize that a vector space is much more than merely a space of vectors, and that it is straightforward and useful to extend it computationally. The crucial idea is to make the subspaces of a vector space explicit elements of computation. To build our algebra of subspaces, we revisit the familiar lines and planes through the origin. We investigate their geometrical properties carefully, and formalize those by the aid of a new algebraic outer product, which algebraically builds subspaces from vectors.

Associativity, distributivity, and antisymmetry then make the outer product of these four vectors zero: a ∧ b ∧ c ∧ d = a ∧ b ∧ c ∧ (α a + β b + γ c) = a ∧ b ∧ c ∧ (α a) + a ∧ b ∧ c ∧ (β b) + a ∧ b ∧ c ∧ (γ c) = 0. So the highest-order element that can exist in the subspace algebra of R3 is a trivector. It should be clear that this is not a limitation of the outer product algebra in general: if the space had more dimensions, the outer product would create the appropriate hypervolumes, each with an attitude, orientation, and magnitude.

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