Theorem-Dimension Of A Vector Space

Theorem:- If W is a subspace of an n-dimensional vector space over the field F then

dimW≤dimV

Proof:- 

Let W be a subspace of a finite dimensional vector space V(F).Let S={α₁,α₂,…………………..α_m} be a basis of V.

∴                       dimV=m

Then      L(S)=V

i.e. each elements of V is a linear combination of elements of S. also            W⊆ V

Hence in particular each elements of W can be generated by linear combination of elements of S. Since S is linearly independent therefore either S is basis of W or any subset of S is a basis of W.

Hence in either cases the basis of W can not have more elements than S.

Consequently

dimW ≤ m

dimW≤dimV