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

• Arnoldjavs3
In summary, the conversation discusses the process of determining if a set is a subspace, using the example of a set M with a specific condition for its vectors. The three aspects needed for a subspace are mentioned, and the process of solving the given problems is discussed. The importance of considering all possible vectors and factors to prove closure under multiplication and addition is emphasized.
Arnoldjavs3

Homework Statement

http://prntscr.com/ej0akz

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
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.

Arnoldjavs3 said:
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##?

Arnoldjavs3
Arnoldjavs3 said:
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.
Arnoldjavs3 said:
That means u1 = 90 or pi/2
Not 90 - don't work in degrees. pi/2 is not the only option, there are more.
Arnoldjavs3 said:
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.
Arnoldjavs3 said:
I'm assuming M contains vector U as well right?
What is U?
Arnoldjavs3 said:
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.

1. How can you determine if a set is a subspace?

To determine if a set is a subspace, you need to check if it follows the three properties of a subspace: closure under addition, closure under scalar multiplication, and containing the zero vector.

2. What is closure under addition?

Closure under addition means that when you add two vectors from the set, the resulting vector is also in the set. Mathematically, if u and v are vectors in the set, then u + v is also in the set.

3. What is closure under scalar multiplication?

Closure under scalar multiplication means that when you multiply a vector from the set by a scalar, the resulting vector is also in the set. Mathematically, if u is a vector in the set and k is a scalar, then ku is also in the set.

4. Why is it important for a subspace to contain the zero vector?

The zero vector is important because it serves as the identity element for addition and scalar multiplication. It also ensures that the set is non-empty and that all subspaces contain at least one vector.

5. Can a subspace have abstract information?

Yes, a subspace can have abstract information. The concept of a subspace can be applied to any set of objects that follow the three properties mentioned above, regardless of the type of information they represent. For example, a subspace of abstract information can be a set of mathematical functions or a set of linguistic structures.

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