I am not entirely sure about the following true/false questions
For all the following : $V$ a vector space and $W_1$ and $W_2$ two subspaces such that $V = W_1 ⊕ W_2$
1) for all subspaces U of V : $U = (U∩W_1)⊕(U∩W_2)$. (i put false here too because i tried with some values.
2) if $B_i$ is a basis of $W_i$ for i =1,2 then $B_1∪B_2$ is a basis of V. (true)
3) If B is a basis of V, then $B∩W_1$ is a basis of $W_1$ (i put true)
Can someone let me know if i am wrong here?
The first is false, consider the plane which is the direct sum of the spaces generated by $(0,1)$ and $(1,0)$. Then the subspace $U=span(1,1)$ is not the sum of the two intersections.
For the second, you have to be careful what you mean by $B_1 + B_2$. In fact, you know that B_1 and B_2 together span the whole $V$, and the cardinality of the set $B_1\cup B_2$ is just right to do that (the generating set is minimal), so $B_1\cup B_2$ is a basis. The set $B_1 + B_2$ is just too big to be a basis!
For the third, consider $W$ as $U$ in my first example, and the basis of $V$ to be the same. If you compute th intersections you find the zero vector, which is not a basis.