Linear transformations
DOWNLOAD
FREE
|
Kernels and Images
Equations are omitted for technical reasons - download the original pdf
The image set of a linear transformation is the sub set of al vectors in the codomain they are values for some vector in the domain. Formally Let [Equation goes here - download the original to see it.] be a linear transformation. The image of[Equation goes here - download the original to see it.], denoted [Equation goes here - download the original to see it.] is the set [Equation goes here - download the original to see it.] Example: Find the images of the transformations [Equation goes here - download the original to see it.] is represented by the matrices A & B of the previous example [Equation goes here - download the original to see it.] An arbitrary element [Equation goes here - download the original to see it.] is [Equation goes here - download the original to see it.] The operation of A on this vector is given by [Equation goes here - download the original to see it.] Since a & b are independent of each other the image of [Equation goes here - download the original to see it. is [Equation goes here - download the original to see it.] = W. The whole of W. An arbitrary element [Equation goes here - download the original to see it.] is [Equation goes here - download the original to see it.] The operation of B on this vector is [Equation goes here - download the original to see it.] So the image set [Equation goes here - download the original to see it.] So the image set is not the whole of V In [Equation goes here - download the original to see it.] is a linear transformation then is a subspace of w. In some cores could be equal to the whole of W. [Diagram goes here - download the original to see it.] The kernel of a linear transformation [Equation goes here - download the original to see it.] is the set of vectors in V that are mapped to the 0 element in W by the transformation [Equation goes here - download the original to see it.] , formally [Equation goes here - download the original to see it.] Example Find the kernel linear transformation represented b the matrix A of the previous example. We require [Equation goes here - download the original to see it.] Hence [Equation goes here - download the original to see it.] Hence a = 0, b = 0 [Equation goes here - download the original to see it.]
|
Contents of
Linear transformations
1 Linear Transformations
2 Finite Linear Transformations
3 Composition of Linear Transformations
4 Kernels and Images
|
