Vector Subspace

A subset of a Vector Space over a field is called a subspace of if is a vector space over under the operations of addition and scalar multiplication defined in

We denote this as ¹

(Frieidberg 1.3) Let be a vector space and a subset of . Then if and only if the following hold:
(Friedberg 1.4) Any intersection of subspaces of is a subspace of .
(Friedberg e1.3.18) Let , then if and only if or
(Friedberg e1.3.20) If and then for any scalars
(Friedberg e1.3.21) If , then is the smallest subspace containing both and
(Friedberg 1.11) If is finite dimensional, then is finite dimensional, and .

Moreover,
- (Friedberg 1.11.1) Any basis for is a subset of a basis in .
A coset for vector spaces can be defined as follows. Let . For any , the set

is called the coset of containing .
- (Friedberg e1.3.29a) if and only if
- (Friedberg e1.3.29c) The following operations are well defined
- (Friedberg e1.3.29d) We can define the Factor Group as the vector space
- (Friedberg e2.4.22) Let be a Linear Transformation. Define the mapping by
  For any coset . Then the following are true:
  - The mapping is well defined.
  - is linear
  - is an isomorphism.
(Friedberg e2.4.15) Let be finite dimensional vector spaces and an isomorphism. Let then

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