Is a subspace still valid without the zero vector?

In summary, a set of vectors is not considered a subspace if it does not contain the zero vector. This is because a subspace must have the property of closure under addition, which is only possible if the zero vector is included. Some definitions also include the requirement of containing the zero vector, while others simply state that the subspace must be non-empty.
  • #1
jeffreylze
44
0
If a set of vectors does not contain the zero vector is it still a subspace?
 
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  • #2
No, because the subspace will have negatives of elements,
i.e., for all v an element of V, (-1)v or -v will be an element.
For the subspace to be closed under addition (a necessary requirement)
v + (-v) = 0 must be an element which implies the zero vector must be in a subspace of vectors.
 
  • #3
Some textbooks include "contains the 0 vector" as part of the definition of "subspace".
Others just say "is non-empty". As DorianG pointed out, if some vector, v, is in the subspace, then so is -v (a subspace is "closed under scalar multiplication") and so is v+ (-v)= 0 (a subspace is "close under addition").
 

FAQ: Is a subspace still valid without the zero vector?

What is a zero vector in a subspace?

A zero vector in a subspace refers to a vector that has all of its components equal to zero. This means that the vector has no direction and no magnitude.

What is the significance of a zero vector in a subspace?

The presence of a zero vector in a subspace is important because it serves as the origin or starting point for all vectors in that subspace. It also helps define the dimension of the subspace and is used in linear algebra operations.

Can a subspace contain more than one zero vector?

No, a subspace can only contain one zero vector. This is because the zero vector is unique and any other vector with all components equal to zero would be considered the same vector.

How is the zero vector related to linear dependence?

The presence of a zero vector in a subspace can indicate that the set of vectors in the subspace is linearly dependent. This means that one or more of the vectors in the subspace can be written as a linear combination of the other vectors.

What happens if a zero vector is removed from a subspace?

If a zero vector is removed from a subspace, the remaining vectors in the subspace will still maintain the same properties and characteristics. However, the dimension of the subspace will decrease by one, as the zero vector is no longer included in the set of vectors.

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