Mathematical Methods for Physics and Engineering - Matematica.NET

MATRICES AND VECTOR SPACESa discussion of how to use these properties to solve systems of linear equations.The application of matrices to the study of oscillations in physical systems istakenupinchapter9.8.1 Vector spacesA set of objects (vectors) a, b, c, ... is said to form a linear vector space V if:(i) the set is closed under commutative and associative addition, so thata + b = b + a, (8.2)(a + b)+c = a +(b + c); (8.3)(ii) the set is closed under multiplication by a scalar (any complex number) toform a new vector λa, the operation being both distributive and associativeso thatλ(a + b) =λa + λb, (8.4)(λ + µ)a = λa + µa, (8.5)λ(µa) =(λµ)a, (8.6)where λ and µ are arbitrary scalars;(iii) there exists a null vector 0 such that a + 0 = a for all a;(iv) multiplication by unity leaves any vector unchanged, i.e. 1 × a = a;(v) all vectors have a corresponding negative vector −a such that a+(−a) =0.It follows from (8.5) with λ = 1 and µ = −1 that−a is the same vector as(−1) × a.We note that if we restrict all scalars to be real then we obtain a real vectorspace (an example of which is our familiar three-dimensional space); otherwise,in general, we obtain a complex vector space. We note that it is common to use theterms ‘vector space’ and ‘space’, instead of the more formal ‘linear vector space’.The span of a set of vectors a, b,...,s is defined as the set of all vectors thatmay be written as a linear sum of the original set, i.e. all vectorsx = αa + βb + ···+ σs (8.7)that result from the infinite number of possible values of the (in general complex)scalars α,β,...,σ.Ifx in (8.7) is equal to 0 for some choice of α,β,...,σ (not allzero), i.e. ifαa + βb + ···+ σs = 0, (8.8)then the set of vectors a, b,...,s, issaidtobelinearly dependent. Insuchasetat least one vector is redundant, since it can be expressed as a linear sum ofthe others. If, however, (8.8) is not satisfied by any set of coefficients (other than242