# Proving Nonempty Sets are Subspaces & Examples

• zxc210188
In summary, the conversation discusses the proof that a nonempty set W is a subspace of a vector space V if and only if the condition "ax+by is in W for all scalars a and b and all vectors x and y in W" is satisfied. The participants also discuss the definition of a subspace and how to show that a set satisfies the conditions for a subspace. They then mention a simple example of two subspaces in a vector space R2 and discuss the challenges of understanding the concept for a foreign student.
zxc210188
(1)Prove that a nonempty set W is a subspce of a vector space V iff a x+b y is an element of W for all scalars a and b and all vectors x and y in W

(2)Give an example showing that the union of two subspaces of a vector space V is not necessarily a subspaces of V

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Have you had any thoughts at all on the problem? Really, (1) nothing more than an exercise in definitions, and you almost have to try in order to get (2) wrong by guessing at an answer.

What is the DEFINITION of "subspace"? Show that if "ax+ by is in W for all numbers a and b and vectors x and y in W" is true then each of the conditions for a subspace are satisfied. Because this is an "if and only if" statement, you need to then turn around and show that, if W satisfies all the conditions for a subspace, then "ax+ by is in W for all numbers a and b and vectors x and y in W" is true.

As for the second part, as Hurkyl said, it's almost impossible to get it wrong! To make it as easy as possible use R2 as V and choose 2 very easy subspaces.

I know it should be prove in two direction, but I have no idea about how to prove it.
I am a student in Taiwan, so my English is not very well. I cannot understand what the textbook talk about compeletely, so I learn it from doing questions and problems, I did try to think on the problems, but I failed.It may be easy for you but hard for a foreign student.

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## 1. What does it mean for a set to be nonempty?

A set is considered nonempty if it contains at least one element. This means that the set is not empty or has no elements.

## 2. How do you prove that a set is a subspace?

To prove that a set is a subspace, you must show that it satisfies the three properties of a subspace: closure under addition, closure under scalar multiplication, and contains the zero vector. This can be done by showing that any two vectors in the set added together or multiplied by a scalar are still within the set.

## 3. Can you give an example of a nonempty set that is also a subspace?

One example of a nonempty set that is also a subspace is the set of all real numbers that are greater than or equal to zero. This set satisfies the three properties of a subspace: any two non-negative real numbers added together or multiplied by a scalar will still result in a non-negative real number, and the set contains the zero vector (the number zero).

## 4. What is the importance of proving that a set is a subspace?

Proving that a set is a subspace is important because it allows us to understand the structure and properties of the set. It also helps us to determine if certain operations or transformations can be applied to the set, and if so, what the resulting set will look like.

## 5. Can a set be both nonempty and not a subspace?

Yes, it is possible for a set to be nonempty but not a subspace. For example, the set of all integers is nonempty, but it is not a subspace because it does not satisfy the closure under scalar multiplication property (multiplying an integer by a non-integer scalar will result in a non-integer, which is not part of the original set).

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