Mathematical Methods for Physics and Engineering - Matematica.NET

MATRICES AND VECTOR SPACESmay be thought of as ‘transforming’ one geometrical entity (i.e. a vector) intoanother.If we now introduce a basis e i , i =1, 2,...,N, into our vector space then theaction of A on each of the basis vectors is to produce a linear combination ofthe latter; this may be written asA e j =N∑A ij e i , (8.23)i=1where A ij is the ith component of the vector A e j in this basis; collectively thenumbers A ij are called the components of the linear operator in the e i -basis. Inthis basis we can express the relation y = A x in component form asy =⎛ ⎞N∑N∑y i e i = A ⎝ x j e j⎠ =i=1and hence, in purely component form, in this basis we havey i =j=1N∑j=1x jN ∑i=1A ij e i ,N∑A ij x j . (8.24)j=1If we had chosen a different basis e ′ i , in which the components of x, y and Aare x ′ i , y′ i and A ′ ij respectively then the geometrical relationship y = A x would berepresented in this new basis byy ′ i =N∑A ′ ijx ′ j.j=1We have so far assumed that the vector y is in the same vector space asx. If,however,y belongs to a different vector space, which may in general beM-dimensional (M ≠ N) then the above analysis needs a slight modification. Byintroducing a basis set f i , i =1, 2,...,M, into the vector space to which y belongswe may generalise (8.23) asA e j =M∑A ij f i ,i=1where the components A ij of the linear operator A relate to both of the bases e jand f i .248