Nonempty Subspace: Proving 0u = 0

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In summary, if we have a subspace that is nonempty and closed under scalar multiplication and vector addition, it follows directly from the defining axioms that ##0\vec{u} = \vec{0}## for every ##\vec{u} \in V##, where ##V## is any vector space. Therefore, there is no need to introduce a subspace to prove this fact. It is simply a consequence of the defining axioms.
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Mr Davis 97
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I have a simple question. Say we have some subspace that is nonempty and closed under scalar multiplication and vector addition. How could we deduce that ##0 \vec{u} = \vec{0}##?
 
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It holds that ##0\vec{u} = \vec{0}## for every ##\vec{u} \in V##, where ##V## is any vector space. This follows directly from the defining axioms and does not require the introduction of a subspace.

What did you try yourself to prove it?
 
  • #3
Mr Davis 97 said:
I have a simple question. Say we have some subspace that is nonempty and closed under scalar multiplication and vector addition. How could we deduce that ##0 \vec{u} = \vec{0}##?

Every subspace is non empty, closed under scalar multiplication and vector addition so no need to say that.

You should show some effort.
 

What is a nonempty subspace?

A nonempty subspace is a subset of a vector space that contains at least one vector and satisfies the subspace properties, including closed under addition and scalar multiplication.

What does it mean to prove 0u = 0 for a nonempty subspace?

Proving 0u = 0 for a nonempty subspace means showing that the zero vector, or the vector containing all zeros, when multiplied by any vector in the subspace, results in the zero vector. This is an important property in linear algebra and is used in various mathematical proofs.

Why is it important to prove 0u = 0 for a nonempty subspace?

Proving 0u = 0 for a nonempty subspace is important because it helps establish the subspace as a true vector space. It also helps in proving other properties and theorems related to vector spaces.

What are some techniques for proving 0u = 0 for a nonempty subspace?

Some techniques for proving 0u = 0 for a nonempty subspace include using the properties of vector addition and scalar multiplication, using mathematical induction, and using direct proof or contradiction. It is also important to understand the definitions and properties of vector spaces.

Can 0u = 0 hold true for a nonempty subspace?

Yes, 0u = 0 can hold true for a nonempty subspace if the zero vector is in the subspace and the subspace satisfies the subspace properties, including closure under addition and scalar multiplication.

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