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* If two vectors u and v are in W, then u+v must also be in W. (This is called "[[closed under addition]]") | * 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]]"). | * 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]]"). | ||
| + | |||
| + | In other words, every linear combination of two elements in W must be contained in W. | ||
Testing these conditions is the best way to see if W is a subspace. | Testing these conditions is the best way to see if W is a subspace. | ||
Revision as of 15:20, 11 March 2013
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").
In other words, every linear combination of two elements in W must be contained in W.
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 $
