Mathematical Physics Vol 1

1.4 Operations on vectors

27

a ′ 1 a ′ 2 a ′ 3 1 0 0 0 1 0 0 0 1

•

a ′

2

a 1 a 2 a 3

a 1

a 2

a ′

1

Table 1.1: Reciprocal bases vectors.

Figure 1.14: Reciprocal vectors in 2D.

1.4.7 Linear dependence of vectors. Dimension of a space Let us now introduce the term linear dependence of a set of vectors a 1 , a 2 , ··· , a n . Definition Vectors a 1 , ··· , a n are linearly dependent if there exist numbers α 1 , ··· , α n , at least one of which is different from zero, such that the following relation holds α 1 a 1 + α 2 a 2 + ··· + α n a n = 0 . (1.26) Conversely, the vectors are linearly independent , if the relation (1.26) is true only when α 1 = α 2 = ··· = α n = 0 , (1.27) Definition A vector space is n – dimensional if it contains n linearly independent vectors, while each system of n + 1 vectors is linearly dependent.

Let us illustrate this with a few examples. Consider two vectors a and b with the same or opposite directions (Fig. 1.15) a

b = k a

Figure 1.15: Collinear vectors.

Then a (real) number k̸ = 0 exists such that:

b = k a ,

(1.28)

and vectors a and b are called collinear vectors. Assuming k = − α

β , the relation (1.28) can be represented as: α a + β b = 0 .

(1.29)