Vector subspace

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]

Related pages

• Parallel projection

References

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