How can you tell if somethign is a subspace with abstract info?

[Arnoldjavs3]

Homework Statement

http://prntscr.com/ej0akz

Homework Equations

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.

 

[PeroK]

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.

To take the first one. For a vector to be in the set $M$ it must have $\sin(u_1) = 1$. Is the zero vector in $M$?

 

[mfb]

I don't understand how this is enough information to find out if they are subspaces.

Which other information do you expect? You have a full definition of the set M. The addition and scalar multiplication follow the usual operations in R⁴.

That means u1 = 90 or pi/2

Not 90 - don't work in degrees. pi/2 is not the only option, there are more.

So if I put in 2u

What do you mean by that?
To check if the zero vector is in M, just check if it satisfies the condition given there.

I'm assuming M contains vector U as well right?

What is U?

But how can I tell if its closed under multiplication and addition if I only know that sin(u1) =1 and nothing else...

Take an arbitrary vector v in your set M, calculate the vector 1/3 v, check if 1/3 v is in M. If it is not, the set is not closed under scalar multiplication. If it is, you have to check other factors or other vectors. To prove that it is closed, you have to check all vectors and all prefactors.