- #1
geoffreythelm
- 11
- 0
Homework Statement
For the following subset W of R3 determine whether or not W is a subspace of R3. If the subset is not a subspace give a specific example to indicate why it is not a subspace.
ii.) W = {(x,y,z): 2x + y + 3z = 0
The Attempt at a Solution
I know how to do this mostly, but there's two bits that I don't understand.
For the 'closed under addition' test, I said if (x1, y1, z1) and (x2, y2, z2) are in W then 2x1 + y1 + 3z1 = 0 and 2x2 + y2 + 3z2 = 0
Thus, 2(x1+x2) + y1 + y2 + 3(z1 + z2) = 0
But how can you just add them together and say they equal zero? Surely you'd have to subtract one from the other? Like x = 0, y = 0 therefore x = y and x - y = 0?
Then, if that is true, (x1 + x2, y1+y2, z1+z2) is in W.
This is the bit I really don't understand. How can you jump from 2(x1+x2) + y1 + y2 + 3(z1 + z2) = 0 to (x1 + x2, y1+y2, z1+z2)?
Thanks!