Question

necessary and sufficient conditions for a non empty subset of a vector space to be a vector subspaces​

Answer 1

Answer:

Theorem: A necessary and sufficient condition for a nonempty subset S of a vector space to be a subspace is “a, b ∈ F, u, v ∈ S ⇒ au + bv ∈ S.” (28) Suppose A ∈ Mm×n(F). Then W = {X ∈ Fn : AX = 0} is a subspace of Fn. This is called the solution space of AX = 0.

Answer 2

Answer:

∀ α, β ∈ W and a, b∈ F

⇒ aα + bβ ∈ W

Step-by-step explanation:

Let V be any vector space over the field F

A non-empty subset W of a vector space V is a subspace of V if and only iff

∀ α, β ∈ W and a, b∈ F

⇒ aα + bβ ∈ W

which is the necessary and sufficient condition for a non-empty subset of a vector space to be a vector subspace