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