Understanding Vector Spaces and Subspaces: Are Vector Spaces Subspaces of Themselves?

Determine whether the statement is True or False. Justify your answer:A vector space is also a subspace of itself.

The statement is true: a vector space is also a subspace of itself

The statement is true: a vector space is also a subspace of itself.

To justify this, we need to understand the definitions of a vector space and a subspace.

A vector space is a set of vectors with two operations, addition and scalar multiplication, that satisfy certain properties. These properties include closure under addition and scalar multiplication, associativity of addition, existence of an additive identity, existence of additive inverses, and distributive properties.

A subspace, on the other hand, is a subset of a vector space that is also a vector space itself. To be a subspace, the subset must satisfy the same properties as the original vector space.

Since a vector space satisfies all the properties of a vector space, it also satisfies the properties of a subspace. Thus, a vector space is a valid subset of itself and can be considered a subspace of itself.

In summary, a vector space is also a subspace of itself because it satisfies all the properties required for a subspace. Therefore, the statement is true.

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