A linear mapping function satisfying:

1. Additivity: .
2. Homogeneity: .

Some mappings hold certain properties:

1. Injective: , . That is, no two elements of the domain map to the same element in the codomain.
2. Surjective: . That is, the range equals the codomain .
3. Bijective: is both injective and surjective. A linear mapping that's bijective is also called an isomorphism, which geometrically preserves structure.

For vector spaces with correspondence 🏹 Vector > Basis and linear mapping , we can construct transformation matrix such that

summarized as

To construct , we find elements satisfying

where is a basis vector in . In other words, each column of expresses the transformed -th basis vector in terms of the resulting basis .

Range and Null Spaces#

The range space (also called image) is the set of all for some input ,

The null space (also called kernel) is the set of all inputs that map to ,

In terms of mappings on vector spaces, we can think of the range space as all vectors that can be reached via and the null space as all vectors that map to . With the transformation matrix , is the set of all linear combinations of and is the set of vectors such that .

Rank and Nullity#

Rank is the dimension of , and nullity is the dimension of . The rank-nullity theorem states that for ,

Intuitively, this is saying that the dimension of the input space can be split into dimensions that map to zero () and dimensions that map to non-zero () elements.

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.