Can a basis for a vector space $V$ can be restricted to a basis for any subspace $W$?
Solution 1:
Take for example $V = \mathbb{R}^2$, $W = \{(x,y)\in\mathbb{R}^2 : x = y\}$, and $K = \{(1,0),(0,1)\}$.
$W$ is a subspace of $V$, but there is no subset of $K$ that gives a basis for $W$.
Solution 2:
A simple counter-example is $V=\mathbb R^2$, $W$ = the $y$-axis, and the basis of $V = \{ (1,0),(1,1)\}$.
Clearly neither of the two linearly independent vectors in this basis lie along the $y$-axis.