🗺️ Linear Mapping

The following are special types of mappings.

1. Injective: if , .
2. Surjective: if .
3. Bijective: if is both injective and surjective.

A mapping is linear if and . If the mapping is linear and bijective, it’s called an isomorphism.

For a basis of , every can be expressed as . Then, are the coordinates of with respect to , and the coordinate vector is

For vector spaces and with corresponding bases and and linear mapping , we can write

Then, the transformation matrix contains these elements . If is the coordinate vector of with respect to and is the coordinate vector of with respect to , then

Using this formulation, we can also define a change of basis matrix that maps basis to within the same vector space. Finally, note that transformation matrices can be composed together via matrix multiplication.

Image and Kernel §

For , the kernel is defined as

and the image is defined as

We can think of the kernel as all vectors that map to and the image as all vectors that can be reached via .

The rank-nullity theorem states that for ,

Affine Spaces and Mappings §

Affine spaces, also known as linear manifolds, are spaces offset from the origin. Thus, they are no longer vector subspaces. Specifically, for vector space , , and subspace , an affine space is defined as

is called direction, and is the support point.

Note that every element can be described as

for basis of . are known as parameters, and this is the parametric equation of .

Following a similar definition, an affine mapping with and is defined as

is the translation vector, and every affine mapping can be seen as a linear mapping followed by a translation.