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| − | In linear algebra, vectors <math>v_1, v_2... v_n</math> form a basis for | + | In linear algebra, vectors <math>v_1, v_2... v_n</math> form a basis for the subspace V when |
| − | *The vectors [[span]] | + | *The vectors [[span]] V. (in other words, the span of the vectors is V) |
*The vectors are [[Linearly Independent|linearly independent]]. | *The vectors are [[Linearly Independent|linearly independent]]. | ||
| − | + | If V is a subspace of <math>\mathbb R^m </math> it follows that n must be less than or equal to m. | |
| − | + | Note that there can be more than one set of vectors that form a basis for the same space. In fact, there are an infinite number of bases (plural of basis) for a subspace provided the subspace is not just <math>\vec 0</math>. However, all bases for a given subspace have the same number of vectors. This number of vectors is called the dimension of the subspace. | |
[[category:MA351]] | [[category:MA351]] | ||
Revision as of 17:07, 4 March 2010
In linear algebra, vectors $ v_1, v_2... v_n $ form a basis for the subspace V when
- The vectors span V. (in other words, the span of the vectors is V)
- The vectors are linearly independent.
If V is a subspace of $ \mathbb R^m $ it follows that n must be less than or equal to m.
Note that there can be more than one set of vectors that form a basis for the same space. In fact, there are an infinite number of bases (plural of basis) for a subspace provided the subspace is not just $ \vec 0 $. However, all bases for a given subspace have the same number of vectors. This number of vectors is called the dimension of the subspace.
