Linear transformations
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Composition of Linear Transformations
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The composition of two linear transformations can be represented by a matrix which is the product of the two matrices representing the individual transformations. If [Equation goes here - download the original to see it.] is a linear transformation with matrix representation A, and if [Equation goes here - download the original to see it.] is a linear transformation with matrix representation B then [Equation goes here - download the original to see it.] is a linear transformation with matrix representation BA. There we state and illustrate this theorem but do not prove it. Example Let V = the set of all quadratic polynomials Let W = the set of all linear polynomials Let [Equation goes here - download the original to see it.] be the operational of taking a derivative Let[ Equation goes here - download the original to see it.] be the operation of the integration Let A be the matrix representation of and let B be the matrix representation of [Equation goes here - download the original to see it.] . Find BA and comment on its form. Solution We have already shown that [Equation goes here - download the original to see it.] with respect to the basis [Equation goes here - download the original to see it.] in V and [Equation goes here - download the original to see it.] in W for [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] Hence The matrix representing [Equation goes here - download the original to see it.] is [Equation goes here - download the original to see it.] However, a word of caution. When integrating [Equation goes here - download the original to see it.] we have to include a constant of integration. [Equation goes here - download the original to see it.] The constant, c, does not depend on a and/or b, or the process itself of taking a derivative but represents the particular solution to the integral [Equation goes here - download the original to see it.] The solution to the equation [Equation goes here - download the original to see it.] is a family of equations each of which shares the common part [Equation goes here - download the original to see it.] , and differs by a constant c representing the particular solution. The particular solution is not part of the process of integration as the reverse differentiation, so the matrix representing integration here is[Equation goes here - download the original to see it.] Hence [Equation goes here - download the original to see it.] BA = I, the identity matrix. This is what is expected since integration and differentiation are reverse processes of each other
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Contents of
Linear transformations
1 Linear Transformations
2 Finite Linear Transformations
3 Composition of Linear Transformations
4 Kernels and Images
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