subspace

(Minerva)

Given , a subset of a vector space .

is a subspace of if it, by itself, is a vector space with the same operations and a field of scalars as

To show that is also a vector space, we only need to show first 3 properties:

1. is closed under
2. is closed under
3. has an additive zero vector

For reference, here’s all properties of a vector space

8 full criteria to prove a vector space

Given a set with two operations (addition and scalar multiplication), is a real-valued vector space if and it

1. Is closed under both operations
2. Sum of vectors is a vector in set

2. Result scalar multiplication is vector in set :

2. has an additive zero vector. It acts as neutral element in a group.

3. has additive inverses for all of its vectors. They act as inverse elements in a group

4. keeps multiplication identity

5. is commutative

6. both and are associative

7. both and are distributive

Link to original

Example

The vector space is the set of all 2x2 matrices and is a subset of containing only 2x2 upper triangular matrix

The vector space is the set of solution to the linear system . Let be a subset of where

Fundamental subspaces

• null space
• row space
• column space