Question

Prove that any subset of V that contain${\mathbf{0}}_{\mathbf{v}}$is linearly dependent over F

Short answer

Answer:

${\mathbf{0}}_{\mathbf{v}}$ is linearly dependent over F.

Step-by-step solution

Definition of space.

A space consisting of vectors, together with the associative and commutative operations of addition of vector and the associative and distributive operation of multiplication of vectors by scalars.

Step 2:Showing that 0v is linearly dependent.

let any subset of V contain $0_v$.

Consider, the set $\left\{0\right\}$be any subset of V that contain $0_v$.

Now if $1\in F$such that

$1·0=0$

But as $1\ne 0$.

Therefore, the set $\left\{0\right\}$is linearly dependent over F.