Tell whether or not the following vectors are linear independent, if they are l.i., say if they generate or and in the contrary case characterize implicitly the subspace generated and give a basis of this subspace.
The vectors are (1,1,2,4),(2,-1,-5,2),(1,-1,-4,0) and (2,1,1,6).
My attempt : I formed and reduced this matrix : to this one . I concluded by saying that the vectors are linear dependent (because of the 2 rows that are 0, so 2 of the 4 vectors are linear dependent between them) and that the vectors generate (because 2 rows are reduced) and that a basis of is .
I have some questions : first, is what I've done correct (at least logically)?
Second question : how can I describe implicitly ?
Third question : is a possible basis of ? What about ? (I guess yes).
Thank you very much.