# If V is a finite dimensional vector space and W

If V is a finite dimensional vector space and W is a subspace, the W is finite dimensional. Prove it.

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David Clayton

By definition-
If V be a vector space over an arbitray field F, then we say that V is finite dimensional if it is spanmed by a finite set of vectors.
Let, dimV =n
$$\displaystyle\Rightarrow$$ V is spamed by a set of n linearly independent vectors in V,
say $$S=\{v_1,v_1,v_1,...,v_n\}$$
Now, as W is a stubspace of then W is spaned by at most n elements of the set S.
Hence, by definition of finite dimensional vector sapace- W is finite dimensional.