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 = { } be a basis for V and B' = { } be a basis for W. Let T: V W be a linear transformation from V into W. Let be the images in W under transformation T of the basis vectors as referred to the W basis B' = { } :

or, in matrix form,

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 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 is the matrix representation of the transformation then for any vector v V

where is the coordinate vector of v as referred to the B basis and 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,

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 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 are 1,x,x², ... ,xⁿ .

 Upon taking the derivatives of the basis vectors

equations (1) above then become

or, in matrix form,

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)

References

  Lipschutz. Linear Algebra. p.156

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