Proving that W1 \cap W2 is a Subspace of V

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Closure under addition: for any two vectors u, v in W, their sum u+v is also in W.2. Closure under scalar multiplication: for any scalar k and any vector u in W, the product ku is also in W.3. Contains the zero vector: the zero vector is always in any subspace.4. Contains all elements of W1 and W2: since W1 and W2 are subspaces, all their elements are also in W.Therefore, W is also a subspace of V, as it satisfies all the necessary conditions.
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mandygirl22
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Prove that if W1 and W2 are subspaces of the vector space V, then W1 [tex]\cap[/tex] W2 is also a subspace of V.

Attempt at solution:
I really don't even know where to start on this because I am confused about how to prove in general that something is a subspace. Also, I don't know how to find what W1 [tex]\cap[/tex] W2 is. Please help!
 
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mandygirl22 said:
Prove that if W1 and W2 are subspaces of the vector space V, then W1 [tex]\cap[/tex] W2 is also a subspace of V.

Attempt at solution:
I really don't even know where to start on this because I am confused about how to prove in general that something is a subspace. Also, I don't know how to find what W1 [tex]\cap[/tex] W2 is. Please help!

W1 [itex]\cap[/itex] W2 consists of all the vectors that are in W1 and in W2.

To make it simpler to write, let's W = W1 [itex]\cap[/itex] W2.

What are the things you need to check to verify that a set is a subspace of a given vector space?
 

1. What is a subspace?

A subspace is a subset of a vector space that satisfies all the properties of a vector space, such as closure under addition and scalar multiplication.

2. How do you prove that W1 \cap W2 is a subspace of V?

To prove that W1 \cap W2 is a subspace of V, we need to show that it satisfies the three properties of a vector space: closure under addition, closure under scalar multiplication, and the existence of a zero vector.

3. Can you give an example of W1 \cap W2 being a subspace of V?

An example of W1 \cap W2 being a subspace of V is the intersection of the x-axis and y-axis in a two-dimensional Cartesian coordinate system. This intersection forms the origin (zero vector) and is closed under addition and scalar multiplication.

4. What is the importance of proving that W1 \cap W2 is a subspace of V?

Proving that W1 \cap W2 is a subspace of V is important because it ensures that this intersection satisfies all the properties of a vector space, which allows us to use the tools and techniques of linear algebra to solve problems related to it.

5. Are there any special cases where W1 \cap W2 may not be a subspace of V?

Yes, there are special cases where W1 \cap W2 may not be a subspace of V. For example, if one of the subspaces W1 or W2 is not a subset of V, then their intersection will also not be a subspace of V. Additionally, if the intersection is empty, then it cannot be considered a subspace.

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