Question

If $\mathit{S}\mathbf{=}\left\{v_1,v_2,.....v_k\right\}$is a linearly dependent subset of V, then prove that any subset of V that contains S is also linearly dependent over F.

Short answer

Answer:

Any subset of V that contains Sis also linearly dependent over.

Step-by-step solution

Definition of linearly dependent vector.

A set of vector is said to be linearly dependent if one of the vector of the set can be written as a linear combinations of the others, and if no vector can be written as a linear combination of others then the vector is said to be linearly independent.

Showing that set of V that contain S is also linearly dependent.

Let S be the linearly dependent subset of vector space Vover a field F.

LetS' be the subset of the vector space V that contain.

SinceS' be the subset of vector space V that contain S.

Therefore, subsetis the super set of S. Since Sis the linearly dependent subset of vector space Vtherefore thee exist a finite subset Tof Ssuch that T is linearly dependent.

As, $T\subseteq S$and $S\subseteq S$

Thus,

$T\subseteq S\subseteq S\text{'}$

This implies that T is also a finite linearly dependent subset ofS'. Therefore, S'is also linearly dependent over F.