Proving aX is in U if X is in Rn and U is Subspace of Rn

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In summary, if aX is in U, where a is a non-zero number and X is in Rn, then X is also in U. This can be shown by the fact that U is closed under scalar multiplication, which is a property of subspaces. Additionally, a non-zero number is important because it allows for scalar multiplication to have a non-trivial effect on the vector X, whereas multiplication by zero would result in a trivial solution.
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stunner5000pt
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Let U be a subspace of Rn

If aX is in U, where a is non zero number and X is in Rn, show that X is in U

THis seems so obvious... but i m not sure how to show this by a proof

aX is in U and aX is in Rn for sure and U is a subspace of Rn.
Is it true that if U is closed under scalar multiplication then X is in U ?

Please advise!
 
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  • #2
Is it true that if U is closed under scalar multiplication then X is in U ?

Yes. And why is U closed under scalar multiplication?
 
  • #3
Muzza said:
Yes. And why is U closed under scalar multiplication?


Ohh, me sir! Is it by any chance to do with the definition of what a subspace is?
 
  • #4
And, by the way, why is it important that a be non-zero? What property do non-zero numbers have that zero does not?
 

FAQ: Proving aX is in U if X is in Rn and U is Subspace of Rn

What is the definition of a subspace?

A subspace is a subset of a vector space that satisfies the three properties of closure under addition, closure under scalar multiplication, and containing the zero vector.

How do you prove that a given set is a subspace?

To prove that a given set is a subspace, you must show that it satisfies the three properties of closure under addition, closure under scalar multiplication, and containing the zero vector.

What is the relationship between a subspace and a vector space?

A subspace is a subset of a vector space. All vector spaces have at least two subspaces, the zero subspace and the entire vector space itself.

What is the difference between a subspace and a span?

A subspace is a subset of a vector space that satisfies the three properties of closure under addition, closure under scalar multiplication, and containing the zero vector. The span of a set of vectors is the set of all possible linear combinations of those vectors.

Can a subspace contain vectors that are not in the original vector space?

No, a subspace must be a subset of the original vector space. All vectors in a subspace must also be in the original vector space.

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