By von Oniscik A.L., Sulanke R.
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The aim of this ebook is to illustrate that advanced numbers and geometry could be combined jointly fantastically. This ends up in effortless proofs and normal generalizations of many theorems in airplane geometry, comparable to the Napoleon theorem, the Ptolemy-Euler theorem, the Simson theorem, and the Morley theorem.
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Extra info for Algebra und Geometrie 2. Moduln und Algebren
1(b). If you think of a line as a set of points, then any scalar multiple of a determines the same line. For many applications, this is enough, but in others you need more detailed properties, such as its heading (opposite for a and −a) and its speed (twice as much for 2a as for a). The characterization of a line by a vector allows it to have those extra properties. 1: Spanning homogeneous subspaces in a 3-D vector space. a (d) c b SPANNING ORIENTED SUBSPACES 26 terms for those features that can transcend vectors.
We list their properties, defining terms that unify those across dimensions. This uncovers an algebraic product that can span them, thus making those subspaces and their properties elements of computation. The concept of a subspace is independent of any metric properties a vector space might have. 2 ORIENTED LINE ELEMENTS 25 defines it. This also implies that we cannot use orthonormal bases in our examples, which may make them look a bit less specific than they could be. Of course, the concepts still work when you do have a metric, and some of the exercises bring this out.
We believe this approach is more accessible than axiomatizing geometric algebra first, and then having to discover its significance afterwards. The book consists of three parts that should be read in order (though sometimes a specialized chapter could be skipped without missing too much). 18 WHY GEOMETRIC ALGEBRA? 1 PART I: GEOMETRIC ALGEBRA First, we get you accustomed to the outer product that spans subspaces (and to the desirability of the “algebraification of geometry”), then to a metric product that extends the usual dot product to these subspaces.