B' is not a basis for , those last 2 elements are the same thus they are clearly not linearly independent.
But for your part a) one you are correct.
I suspect something has been copied down incorrectly one place or another.
In any case, the idea is correct for your part b so far. What you need to do is see what the image of the first basis element is as you have done, it is (k,0,0). Then see how to represent it in terms of the other basis. That is precisely the problem, that B' set does not span so this will not be possible in this case and this is why you are having trouble here.