please solve clealy and step by step Solution a The set A is

please solve clealy and step by step.

Solution. (a) The set A is not a subspace, since it does not even contain 0 (Indeed 0 E A would mean that 0 2, which is absurd.) (b) The set B is a subspace. Indeed, it is easy to check all three axioms. (c) The set C is a subspace. Indeed, x 0 is equivalent to x1 0, and thus C is simply (x1,x2, 3)TER3 x1 0 which is easily seen to be a subspace (the proof is just like that for part (b), but simpler) (d) The set D is not a subspace. Indeed, it violates axiom (b), because if we set v (1,1,1) and w (1,1, 1) then v E D and w E D (namely, v w u, u 2 u 3) T for u 1), but v w D (indeed, v w (2,2, 2) does not have the form (u, u2, u 3) for any u E R, because any such u E IR would have to satisfy u 2 and u 2 and u 2 at the same time, which is impossible) (It also violates axiom (c); but of course, it is enough to find one contradiction.) (e) The set E is a subspace. It can be rewritten as follows (1 u, 0) u E IR (1+u) (1,-1,0) v (1, 1,0) I v E R (here, we have substituted v for 1 u) (1, -1,0)

Solution

You posted answers not question. Please attach question as a comment, or mail to a.sudhirk@yahoo.com. I will explain you very clearly. If you want repost it.