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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}:


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.


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



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


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


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):


Suppose now that V is the vector space of polynomials


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


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

 Upon taking the derivatives of the basis vectors






equations (1) above then become



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)



  Lipschutz. Linear Algebra. p.156

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

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