If B is a mxn matrix and V the solution space of the homogeneous linear system Bx=0, and there exists a set of linearly independent vectors $u_1$, $u_2$,...,$u_k$ in $R^n$,  Prove that if V intersects span{$u_1$, $u_2$, ..., $u_k$} = {0}, then $Bu_1$, $Bu_2$,..., $Bu_k$ are linearly independent.  And if V intersects span{$u_1$, $u_2$, ..., $u_k$} is not equals to {0}, are $Bu_1$, $Bu_2$, ..., $Bu_k$ linearly independent?
I know that any subset of V with more than k vectors is always linearly dependent, but how do I include matrix A into the vectors given to prove the statement?   Also, for part (b), does the statement mean that Ax=0 only has the trivial solution since V intersects span{u1, u2, ..., uk}={0}?  Any help would be appreciated! Thank you!
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