Given $\displaystyle u_1 , u_2 , u_3 , u_4$ is a basis for $\displaystyle R^4$.

$\displaystyle v_1=ku_1-u_3+u_4$ , $\displaystyle v_2=u_1+u_2-u_4$ , $\displaystyle v_3=4u_2+ku_3-6u_4$

a) For which values of k are vectors $\displaystyle v_1, v_2, v_3$ 1) independent 2) dependent

b) In cases where $\displaystyle v_1, v_2, v_3$ are dependent, write $\displaystyle v_3$ as a linear combination of $\displaystyle v_1$ and $\displaystyle v_2$

c) For which values of k does the group $\displaystyle u_1, u_2, u_3, v_1, v_2$ span $\displaystyle R^4$

a) I proved that when k does not equal 2, they are independent. And when k = 2 they are dependent.

b) i proved that $\displaystyle -2v_1+4v_2=v_3$

c) and this is where i am having trouble. I came to the conclusion that the group spans $\displaystyle R^4$ as long as $\displaystyle v_1$and$\displaystyle v_2$ both are not linear combination of $\displaystyle u_1, u_2, u_3$. (It doesn't matter if one of the two is a linear combination since there are 5 vectors) It is given that $\displaystyle u_1 , u_2 , u_3 , u_4$ is a basis for $\displaystyle R^4$. That means that $\displaystyle u_1 , u_2 , u_3$ are independent of each other. By doing the work, I got that $\displaystyle v_1$and$\displaystyle v_2$ are both not linear combinations of $\displaystyle u_1 , u_2 , u_3$ which means it doesn't matter what value $\displaystyle k$ is. Can someone please help me with this. Thank you very much