1. The problem statement, all variables and given/known data Are the following sets subspaces of R3? The set of all vectors of the form (a,b,c), where 1. a + b + c = 0 2. ab = 0 3. ab = ac 2. Relevant equations Each is its own condition. 1, 2 and 3 do not all apply simultaneously - they're each a separate question. 3. The attempt at a solution I know from the back of the book that 1 is a subspace, while 2 and 3 are not. I know that the subset must be closed under addition and scalar multiplication for it to be a subspace, but I'm having trouble how seeing the conditions listed make (a,b,c) a subspace or not. Here's me guessing 1. Who cares if they all equal zero when added together? (a,b,c)+(d,e,f) = (a+d,b+e,c+f), so we're fine. k(a,b,c) = (ka,kb,kc), so we're good. 2. a or b must equal zero. So we're left with (0,b,c) for example. Adding something of the form (a,b,c) could result in a non-zero in a, so we're not closed under addition, making us not a subspace. Correct? 3. b is equal to c. So we actually have something of the form (a,b,b) but we can add (d,e,f), and the sum does not necessarily leave us with b+e=b+f. Am I right or totally messing this up?