A basis of a subspace is subset of a basis of the whole space

Question

If $X$ is a vector space with a basis $B$ and $A$ is a subspace of $X$. Does $A $always has a basis subset of $B$? If yes, how should I prove this?

If no, we should give an example of a vector space $X$ with a basis $B$ and a subspace $A$ of $X$ such that any basis of $A$ is not subset of $B$.

score 11 · Answer 1 · answered Apr 24 '14 at 13:25

11

No. Look at the plane $\mathbb R^2$ with the usual basis vectors $(1,0)$ and $(0,1)$ and the subspace $A = \{(x,x) : x \in \mathbb R\}$.

answered Apr 24 '14 at 13:25

Umberto P.

Thanks so what if both the space and the subspace have the same dimension? – Sara Apr 24 '14 at 14:50
${(1,1), (-1,1)}$ is a basis of the plane. – Umberto P. Apr 24 '14 at 14:53
I mean, What if we repeat the same question with extra condition that both X and A are n-dimensional ? – Sara Apr 24 '14 at 15:01
@Sara if space and subspace have the same dimension then, space and subspace are equal – manifold Mar 07 '18 at 13:50

1 Answers1