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* 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]]"). | ||
| − | Testing these conditions is the best way to see if | + | Testing these conditions is the best way to see if W is a subspace. |
| − | + | Some common subspaces of <math>{\mathbb R}^3</math> | |
| + | *The zero vector, <math> \vec 0 </math> | ||
| + | *A line running through the origin | ||
| + | *A plane passing through the origin | ||
| + | *<math>{\mathbb R}^3</math> | ||
[[Category:MA351]] | [[Category:MA351]] | ||
Revision as of 15:59, 4 March 2010
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 $
