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Let be a linear transformation for all and :
In general, linear transformations are of the form
If is a transformation from to , then will be a matrix.
Let be a linear transformation from 2D space to 1D space.
The following transformation is not linear because it fails on (1) above:
such that .
Image and Kernel
- Set of vectors in such that
- Very analogous to null space of a matrix.
- Set of vectors in such that for some vector in
- Subspace of will have image contained in
Let be a linear transformation, and be a subspace. Then
- The kernel of is a subspace of
- is a subspace of . In particular, is a subspace of
Let be defined as follows:
Kernel is .
, therefore is the kernel of .
For a subspace , The image .
Matrix Representations of Linear Transformations
Let be a matrix, and . Then
is called the standard matrix representation of .
If is a linear transformation mapping into , there is a matrix such that for each . In fact, the th column vector of is given by
. Find the standard matrix representation.