# Maths - Vector Subspaces

For 'H' to be a subspace of a vector space 'V' it must have 3 properties:

• The zero vector of V is in H.
• For each 'u' and 'v' in H then v+u is also in H (closed under +).
• For each 'v' in H and scalar 's' then s*v is also in H (closed under scalar multipication).

### Example of a Subspace

V
 x y z
a vector space in 3 dimensions x,y and z
H
 x y 0
a subspace of V formed by setting the z dimenstion to zero.

### Theorm

If v1…vp are in a vector space V, then span{v1…vp}is a subspace of V.

### Lattice of Vector Subspaces

We can represent the relationships between a whole set of subspaces by a lattice structure (lattices are described on this page). One way to encode this lattice structure of algebras is, as a Clifford algebra, as explained on this page.

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