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| + | =What is a "subspace" in linear algebra?= | ||
A subset (call it W) of vectors is a subspace when it satisfies these conditions: | A subset (call it W) of vectors is a subspace when it satisfies these conditions: | ||
* W contains the [[zero vector]] | * W contains the [[zero vector]] | ||
| − | * If two vectors u and v are in W, then | + | * 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 vectors in W is also in W. | ||
| − | 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> | ||
| + | ---- | ||
| + | [[Linear_Algebra_Resource|Back to Linear Algebra Resource]] | ||
| + | |||
| + | [[MA351|Back to MA351]] | ||
[[Category:MA351]] | [[Category:MA351]] | ||
Latest revision as of 15:27, 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 vectors in W is also 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 $
