1. The problem statement, all variables and given/known data http://prntscr.com/ej0akz 2. Relevant equations 3. The attempt at a solution I know there are three problems in one here, but they are all of the same nature. I don't understand how this is enough information to find out if they are subspaces. It's all really abstract to me. I know that you need three aspects to be a subspace: 1. Must contain zero vector 2. Closed under addition 3. Closed under scalar multiplication. So how can I use this info to solve these problems? For example, in question 40 it tells you that the vector U inside of r4 has the condition that sin(u1) = 1. That means u1 = 90 or pi/2. So if I put in 2u, does that indicate i don't get the zero vector? I'm assuming M contains vector U as well right? But how can I tell if its closed under multiplication and addition if I only know that sin(u1) =1 and nothing else... There are 6 problems of this nature in my textbook that I am unable to solve and they do not give ample explanations for them.