MATH 323 Lecture 16

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Linear Transformations

Let be a linear transformation for all and :

In general, linear transformations are of the form


Transdimensional Transformations

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:

Identity Transformation

such that .

Image and Kernel

Giver ,

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

Theorem 4.1.1

Let be a linear transformation, and be a subspace. Then

  1. The kernel of is a subspace of
  2. 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 .

Theorem 4.2.1

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.