Linear algebra. Find a counter-example

Question

this is the statement: if $\vec v_{1}, \vec v_{2} , \vec v_{3}, \vec v_{4}$ is a basis for the vector space $\Bbb R^{4} $, and W is a subspace of $\Bbb R^{4}$, then some subset of the $\vec v$ 's is a basis for W. Now I need to found a counter-example for this statement.

Answer 1

Try using the standard basis $\vec{v_i}=\vec{e_i}$, and let $W$ be a straight line which doesn't coincide with any of the 4 axes.

Answer 2

Hint Consider $W=span(v_1+v_2)$.

Answer 3

Just take any two linearly-indepedent vectors in $\mathbb R^4$ that are neither of $v_1, v_2, v_3, v_4$.

EDIT: take any two vectors neither of which is a linear combination of {$v_1, v_2, v_3, v_4$}. Alternatively, consider the subspace generated by any pair of vectors {$v_1, v_2, v_3, v_4$} and rotate it about the origin by a non-zero angle, so that it remains a subspace, and so that it does not agree with the planes generated by the other vectors . Or take any pair of vectors, consider the lines generated by these vectors, then rotate the lines (about the origin, so that the lines pass thru the origin), so that the rotated lines do not agree with the remaining vectors in the basis, and consider the plane generated by the rotated lines.