A vector subspace is a vector space that is a subset of another vector space. This means that all the properties of a vector space are satisfied. Let W be a non empty subset of a vector space V, then, W is a vector subspace if and only if the next 3 conditions are satisfied:^[1][2]

1. additive identity – the element 0 is an element of W: 0 ∈ W
2. closed under addition – if x and y are elements of W, then x + y is also in W: x, y ∈ W implies x + y ∈ W
3. closed under scalar multiplication – if c is an element of a field K and x is in W, then cx is in W: c ∈ K and x ∈ W implies cx ∈ W.

If 👁 {\displaystyle W_{1}}
and 👁 {\displaystyle W_{2}}
are subspaces of a vector space 👁 {\displaystyle V}
, then the sum and the direct sum of 👁 {\displaystyle W_{1}}
and 👁 {\displaystyle W_{2}}
, denoted respectively by 👁 {\displaystyle W_{1}+W_{2}}
and 👁 {\displaystyle W_{1}\oplus W_{2}}
,^[3] are subspaces as well.^[4]

• Parallel projection