What are the two closure properties required for a nonempty subset of a vector space to be a subspace?

Conditions

• The subset must be nonempty.
• The subset must be closed under vector addition.
• The subset must be closed under scalar multiplication.

Reasoning, step by step

1. Verify the subset is nonempty.
2. Check if adding any two vectors in the subset results in a vector still in the subset.
3. Check if multiplying any vector in the subset by any scalar results in a vector still in the subset.

Example

Vectors of the form (x,0,−x) stay in the same form after addition or scalar multiplication.

Common misconceptions

• Believing that a subset containing the zero vector is automatically a subspace without checking closure.
• Thinking that closure under addition alone is sufficient without scalar multiplication.

Watch the explanation

Explore next

• How do you verify that a specific set of vectors is closed under addition?
• Why does the nonempty requirement ensure the zero vector is present?