A subset W of a vector space V is a subspace if (1) W is non-empty (2) For every ¯v, ¯w ∈ W and a, b ∈ F, a¯v + b ¯w ∈ W. are called linear combinations. So a non-empty subset of V is a subspace if it is closed under linear combinations.
How do you know if a subset is a subspace?
To show a subset is a subspace, you need to show three things:
- Show it is closed under addition.
- Show it is closed under scalar multiplication.
- Show that the vector 0 is in the subset.
How do you know if it's a subspace?
In other words, to test if a set is a subspace of a Vector Space, you only need to check if it closed under addition and scalar multiplication. Easy! ex. Test whether or not the plane 2x + 4y + 3z = 0 is a subspace of R3.
