Linear transformations
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Finite Linear Transformations
We now consider the case above where [Equation goes here - download the original to see it.]is a linear transformation between finite - dimensional vector spaces. We can give [Equation goes here - download the original to see it.] a matrix representation. So let [Equation goes here - download the original to see it.] w are vector spaces, a linear transformation let [Equation goes here - download the original to see it.] be a basis for V and let[Equation goes here - download the original to see it.] be a basis for W. Since[Equation goes here - download the original to see it.]is a linear transformation [Equation goes here - download the original to see it.] lies in W, and hence [Equation goes here - download the original to see it.] where [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] The numbers [Equation goes here - download the original to see it.] can be arranged into a rectangular matrix thus [Equation goes here - download the original to see it.] If we operate with this matrix on the column vector representing the basis [Equation goes here - download the original to see it.] of V we obtain the basis [Equation goes here - download the original to see it.] of W [Equation goes here - download the original to see it.] Example Let V = set of all quadratic polynomials and W = set of all linear polynomials. Let [Equation goes here - download the original to see it.] be the linear transformation corresponding to the operation of taking a derivative Find the matrix representing [Equation goes here - download the original to see it.]and use it to differentiate [Equation goes here - download the original to see it.] Solution Polynomials in V take the form and w has standard basis [Equation goes here - download the original to see it.] Let D be the linear transformation corresponding to taking the derivative. Then: [Equation goes here - download the original to see it.] Hence the matrix representing A is [Equation goes here - download the original to see it.] The polynomial [Equation goes here - download the original to see it.] has coordinates (3, -2, 4) with respect to the basis [Equation goes here - download the original to see it.] hence [Equation goes here - download the original to see it.] is equivalent to the matrix operation [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.]
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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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