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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 = {α_{1}, α_{2}, .... , α_{n}} be a basis for V and B' = {β_{1}, β_{2},
.... , β_{n}} be a basis for W. Let T: V →W be a linear transformation from V into W. Let α_{1}', α_{2}', ....
, α_{n}' be the images in W under transformation T of the basis vectors α_{1}, α_{2}, .... , α_{n} as referred to
the W basis B' ={β_{1}, β_{2}, .... , β_{n}}:

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 A_{B,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 A_{B,B'} is the matrix representation of the
transformation, then for any vector v
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, x^{2}, ... , x^{n}} 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, x^{2}, ... , x^{n} 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 α_{1}, α_{2}, .... , α_{n} are 1, x, x^{2}, ... , x^{n} .

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

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