Abstract
In this section we continue to use the notation of Chaps. 3–5 and write K for an arbitrary commutative ring with unit and \(\mathbb{k}\) for an arbitrary field.
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Notes
- 1.
Or K- linear, if the precise reference on the ring of scalars is important.
- 2.
See Sect. 2.6 on p. 31.
- 3.
Also collinear or linearly related.
- 4.
Or linearly generates.
- 5.
See Example 6.3 on p. 124.
- 6.
More scientifically, we could say that K n is a direct product of n copies of the abelian group K equipped with componentwise multiplication by elements λ ∈ K.
- 7.
To save space, we shall usually write them in rows. However, when the column notation becomes more convenient, we shall use it as well.
- 8.
See Sect. 6.2.4 on p. 134.
- 9.
Since if v 1 = 0, for instance, then we can take λ 1 = 1 and all the other λ i = 0.
- 10.
See Sect. 6.1.3 on p. 125.
- 11.
See formula (6.14) on p. 128.
- 12.
See Definition 1.2 on p. 16.
- 13.
It asserts that if there are injective maps of sets A ↪ B and B ↪ A, then there is a bijection \(A\stackrel{\sim }{\longrightarrow }B\).
- 14.
See Sect. 6.1.2 on p. 124.
- 15.
See Example 6.6 on p. 128.
- 16.
See Sect. 2.5 on p. 30.
- 17.
Compare with Example 6.10 on p. 130.
- 18.
Also called a parallel displacement .
- 19.
See Sect. 1.3.4 on p. 12.
- 20.
Or center of gravity.
- 21.
Each pair of coinciding points p i = q j appears in the total collection with weight λ i +μ j .
- 22.
Each collection contains k vectors and is obtained by removing either the vector e ℓ+2 or the vector e ℓ+1 from the collection e 1, e 2, …, e k+1 of k + 1 vectors.
- 23.
See Sect. 5.2 on p. 106.
- 24.
By definition, the total degree of a monomial \(x_{1}^{\alpha _{1}}x_{2}^{\alpha _{2}}\,\cdots \,x_{m}^{\alpha _{m}}\) is equal to α 1 +α 2 + ⋯ +α m . The total degree of a polynomial f is defined as the maximal total degree of the monomials in F.
- 25.
A polynomial f(x 1, x 2, …, x m ) is symmetric if \(f(x_{g_{1}},x_{g_{2}},\ldots,x_{g_{m}}) = f(x_{1},x_{2},\ldots,x_{m})\) for every permutation g = (g 1, g 2, …, g m ) ∈ S m . For example, the polynomial (x 1 − x 2)2(x 1 − x 3)2(x 2 − x 3)2 is symmetric, whereas the polynomial (x 1 − x 2)(x 1 − x 3)(x 2 − x 3) is not.
- 26.
That is, such that F 2 = F but F ≠ 0 and F ≠ Id V .
- 27.
That is, such that F 2 = Id V but F ≠ Id V .
- 28.
We say that a point c divides the segment [a, b] in the ratio α: β if \(\beta \cdot \overrightarrow{ca} +\alpha \cdot \overrightarrow{cb} = 0\).
- 29.
A point is the midpoint of a segment if that point divides the segment in the ratio 1: 1.
- 30.
A homothety with center \(c \in \mathbb{A}\) and ratio \(\lambda \in \mathbb{k}\) is a map \(\gamma _{c,\lambda }: \mathbb{A}^{n} \rightarrow \mathbb{A}^{n}\), \(p\mapsto c +\lambda \overrightarrow{ cp}\).
References
Bernoulli, J.: Ars conjectandi, opus posthumum. Accedit Tractatus de seriebus infinitis, et epistola gallicé scripta de ludo pilae reticularis. Basileae, impensis Thurnisiorum, fratrum (1713).
Borevich, Z. I., Shafarevich, I. R.: Number Theory. Academic Press, New York (1966).
Danilov, V. I., Shokurov, V. V.: “Algebraic Curves, Algebraic Manifolds and Schemes.” In Encyclopedia of Mathematical Sciences. Springer, Heidelberg (1994).
Francis, G. K.: A Topological Picturebook. Springer, Heidelberg (1987).
Gorenstein, D., Lyons, R. Solomon, R.: The Classification of the Finite Simple Groups. Mathematical Surveys and Monographs 40, vols. 1–6. AMS Providence, R.I. (1994–2005).
Humphreys, J. E.: Linear Algebraic Groups. Springer, Heidelberg (1975).
Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory. Springer, Heidelberg (1990).
Mumford, D.: Tata Lectures on Theta I. Progress in Math, vol. 23, Birkhäuser (1983).
Serre, J.-P.: Lie Groups and Lie Algebras: 1964 Lectures Given at Harvard University. Lecture Notes in Mathematics series 2, vol. 1500, Springer (1965).
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Gorodentsev, A.L. (2016). Vectors. In: Algebra I. Springer, Cham. https://doi.org/10.1007/978-3-319-45285-2_6
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