U as Subspace of V & W: True or False?

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In summary, if U is a subspace of V, and V is a subspace of W, then U is also a subspace of W. This can be proven by showing that U satisfies the definition of a subspace and that it is a subset of W. There is no example that can disprove this statement, as the logic is sound.
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1. True or False: If U is a subspace of V, and V is a subspace of W, U is a subspace of W.

If true give proof of answer, if false, give an example disproving the statement.


2. My thoughts: If U is a subspace of V, then the zero vector is in V. As well as x+v is in V and ax is in V (by definition of a subspace). If these three are in V, and V is in W, then these three must be in W as well. Therefore U will be a subspace of W. However, I don't know if there is an example to disprove this, or if my logic is completely flawed.

Thanks for any help!
 
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U is a subset of V, and V is a subset of W. This implies that U is a subset of W. Since U is closed under addition and scalar multiplication, we conclude...
 

FAQ: U as Subspace of V & W: True or False?

What does it mean for U to be a subspace of V and W?

For U to be a subspace of V and W, it means that U is a subset of both V and W and also satisfies the properties of a vector space. This means that U must contain the zero vector, be closed under addition and scalar multiplication, and must contain all linear combinations of its vectors.

Is it possible for U to be a subspace of V and W at the same time?

Yes, it is possible for U to be a subspace of both V and W at the same time. This occurs when U is a subset of both V and W and also satisfies the properties of a vector space.

What happens if U is not a subspace of V and W?

If U is not a subspace of V and W, then it is not a subset of both V and W or it does not satisfy the properties of a vector space. In this case, U would not be considered a subspace of V and W.

Can U be a subspace of V and W if V and W are not subspaces of each other?

Yes, U can still be a subspace of V and W even if V and W are not subspaces of each other. This is because the properties of a subspace only require U to be a subset of both V and W, not for V and W to be subspaces of each other.

What is the importance of U being a subspace of V and W?

The importance of U being a subspace of V and W is that it allows for the study of vector spaces within a larger space. This can be useful in various mathematical and scientific fields, such as linear algebra, physics, and engineering.

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