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,…………………..αm}
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
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