Basis Vector

3 Key excerpts on "Basis Vector"

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  Elementary Linear Algebra

  • Howard Anton, Anton Kaul(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  We will refer to these as the “basis vec- tors” for the coordinate system. In summary, it is the directions of the Basis Vectors that establish the positive directions, and it is the lengths of the Basis Vectors that establish the spacing between the integer points on the axes (Figure 4.5.4). 240 CHAPTER 4 General Vector Spaces y x 1 2 –2 –1 1 2 3 –3 –2 –1 y x 2 –2 1 2 3 –3 –2 –1 –3 –4 3 4 1 –1 y y x 1 2 –2 –1 1 2 3 –3 –2 –1 x 2 –2 1 2 3 –3 –2 –1 –3 –4 3 4 1 –1 Equal spacing Perpendicular axes Unequal spacing Perpendicular axes Equal spacing Skew axes Unequal spacing Skew axes FIGURE 4.5.4 Basis for a Vector Space Our next goal is to extend the concepts of “Basis Vectors” and “coordinate systems” to general vector spaces, and for that purpose we will need some definitions. Vector spaces fall into two categories: A vector space  is said to be finite-dimensional if there is a finite set of vectors in  that spans  and is said to be infinite-dimensional if no such set exists. Definition 1 If  = {v 1 , v 2 , . . . , v n } is a set of vectors in a finite-dimensional vector space , then  is called a basis for  if: (a)  spans  . (b)  is linearly independent. If you think of a basis as describing a coordinate system for a finite-dimensional vec- tor space , then part (a) of this definition guarantees that there are enough Basis Vectors to provide coordinates for all vectors in , and part (b) guarantees that there is no interre- lationship between the Basis Vectors. Here are some examples. EXAMPLE 1 | The Standard Basis for R n Recall from Example 1 of Section 4.3 that the standard unit vectors e 1 = (1, 0, 0, . . . , 0), e 2 = (0, 1, 0, . . . , 0), . . . , e n = (0, 0, 0, . . . , 1) span  n and from Example 1 of Section 4.4 that they are linearly independent. Thus, they form a basis for  n that we call the standard basis for R n .

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  Linearity and the Mathematics of Several Variables

  • Stephen A Fulling, Michael N Sinyakov;Sergei V Tishchenko;;(Authors)
  • 2000(Publication Date)
  • WSPC

    (Publisher)

  The possibility of doing this meaningfully depends on regarding the geometry of £-space (in particular, its dot product — see Chapter 6) as having a direct physical significance. 4.2. Bases from coordinate systems 167 The drawing shows this entire assemblage of vectors at two points, one of which has r — 1 so that all three ^-related vectors are the same. Most of the coordinate systems commonly used are orthogonal — mean-ing that the coordinate curves (u* varying, all other us constant) are perpen-dicular to the coordinate surfaces (uk constant, all other us varying). Thus dk and cjt are parallel in such a case. In fact, we see in the polar example that their lengths are reciprocal. (This makes sense: If a unit change in 9 produces a large change in x , then a unit change in a component of x ought to produce a small change in 6; and this scaling factor clearly is controlled by r.) The inverse function theorem treated in Sec. 5.5 shows that this property holds for all orthogonal coordinate systems. For a nonorthogonal coordinate system, however, the two associated sets of Basis Vectors at each point won't be parallel. For example, let x — u + v, y = v. The inverse transformation is u = x — u, v = y. We easily calculate, for instance, *-(§)-(!)■ *■(!)-(?)• We note (see drawing) that ek is perpendicular to c and that di is perpen-dicular to

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  Modern General Relativity

  Black Holes, Gravitational Waves, and Cosmology

  • Mike Guidry(Author)
  • 2019(Publication Date)
  • Cambridge University Press

    (Publisher)

  (2.3) For example, in terms of the spherical coordinates (r , θ , ϕ), r = (r sin θ cos ϕ)i + (r sin θ sin ϕ)j + (r cos θ )k. (2.4) The second coordinate system in these examples generally is not cartesian but the space still is assumed to be intrinsically euclidean (not curved). The transformation from the (x , y , z) coordinates to the (r , θ , ϕ) coordinates just gives two different schemes to label points in the same flat space. This distinction is important because shortly we will generalize this discussion to consider coordinate transformations in spaces that are intrinsically curved (non-euclidean geometry). 2.1.2 Basis Vectors Vectors are geometrical objects: they have an abstract mathematical existence independent of their representation in any particular coordinate system. However, it is often of practical utility to express vectors in terms of components within a specific coordinate system by defining a set of Basis Vectors that permit arbitrary vectors to be expanded in terms of that basis. Equations (2.3) and (2.4) are familiar examples, where an arbitrary vector has been expanded in terms of the three orthogonal cartesian unit vectors (i , j , k) pointing in the x , y , and z directions, respectively. If the space is flat, a single universal basis can be chosen that applies to all points in the space [for example, the cartesian unit vectors (i , j , k)]. If the space is curved, as will be the case in general relativity, it is no longer possible to define a single coordinate system valid for the whole space and it is often useful to define Basis Vectors at individual points of the space, as we shall now describe. Parameterized curves and surfaces: At any point P (u 0 , v 0 , w 0 ) defined at specific coordinates (u 0 , v 0 , w 0 ) of a space, three surfaces pass. They are defined by setting u = u 0 , v = v 0 , or w = w 0 , respectively. The intersections of these three surfaces define three curves passing through P (u 0 , v 0 , w 0 ).

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