What is a "subspace" in linear algebra?

A subset (call it W) of vectors is a subspace when it satisfies these conditions:

• W contains the zero vector
• If two vectors u and v are in W, then u+v must also be in W. (This is called "closed under addition")
• If the vector v is in W, and k is some scalar (ie just some number), then kv must also be in W. (This is called "closed under scalar multiplication").

Testing these conditions is the best way to see if W is a subspace.

Some common subspaces of $ {\mathbb R}^3 $

• The zero vector, $ \vec 0 $
• A line running through the origin
• A plane passing through the origin
• $ {\mathbb R}^3 $

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