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MATRIX REPRESENTATION OF A LINEAR TRANSFORMATION      


 

Let V be some abstract n-dimensional vector space over field F and let W be an abstract m-dimensional vector space over field F. Let B = {α1, α2, .... , αn} be a basis for V and B' = {β1, β2, .... , βn} be a basis for W. Let T: V →W be a linear transformation from V into W. Let α1', α2', .... , αn' be the images in W under transformation T of the basis vectors α1, α2, .... , αn as referred to the W basis B' ={β1, β2, .... , βn}:



ole.gif  



or, in matrix form,




ole1.gif     ole2.gif ole3.gif                                                                                                                                                



The matrix representation of T relative to the bases B and B' is given by the transpose of the above matrix of coefficients i.e.



                ole4.gif  



The i-th column of mxn matrix AB,B' consists of the coordinates in W space of the image of the i-th basis vector with respect to the B' basis.


Theorem. If T is a linear transformation T: V → W from an abstract n-dimensional space V to an m-dimensional space W and mxn matrix AB,B' is the matrix representation of the transformation, then for any vector v ole5.gif V

 

                 ole6.gif  


where [v]B is the coordinate vector of v as referred to the B basis and [T(v)]B' is the coordinate vector of the image of v (in W) as referred to the B' basis.



Thus if we multiply the coordinate vector of v by the matrix representation of T, we obtain the coordinate vector of the image of v.



A special case of the above is the case where we map an n-dimensional vector space V onto itself, T: V →V and B’ = B. In this case the matrix representation of T is n-square.


 


 


Example. Let V be the space of all polynomials


                         ole7.gif


of degree ≤m over the field F of real numbers and let B = {1, x, x2, ... , xn} be a basis for the space. Let T: V →V be a linear mapping on V, The images of the n = m + 1 basis vectors

  1, x, x2, ... , xn are given by



ole8.gif                                                                                                                                                  




or, in matrix form,



ole9.gif    ole10.gif ole11.gif




The matrix representation of T relative to the basis B is then given by the transpose of the matrix of coefficients of (4):




             ole12.gif




Suppose now that V is the vector space of polynomials


              ole13.gif


of degree ole14.gif 3 over field R of real numbers and suppose that the linear mapping T is that of the differential operator D defined by



                          ole15.gif



The basis vectors α1, α2, .... , αn are 1, x, x2, ... , xn .


 Upon taking the derivatives of the basis vectors


     

ole16.gif     

ole17.gif        

ole18.gif      

ole19.gif



equations (1) above then become


  

ole20.gif  



or, in matrix form,



ole21.gif     ole22.gif ole23.gif



The matrix representation of the linear transformation effected by the differential operator D relative to the basis B is then given by the transpose of the matrix of coefficients in (6)



ole24.gif  





References

  Lipschutz. Linear Algebra. p.156

  Hoffman, Kunze. Linear Algebra. p. 79,80



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