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=='''Basis'''== | =='''Basis'''== | ||
| − | The vectors v1, v2,..., vk in a vector space V are said to form a '''basis''' for V if (a) v1, v2,..., vk span V and (b) v1, v2,..., vk are linearly independent. | + | '''Definition:''' The vectors v1, v2,..., vk in a vector space V are said to form a '''basis''' for V if (a) v1, v2,..., vk span V and (b) v1, v2,..., vk are linearly independent. |
| + | ===Note*=== | ||
| + | If v1, v2,..., vk form a basis for a vector space V, then they must be distinct and nonzero. | ||
| + | ===Note**=== | ||
| + | The above definition not only applies to a finite set of vectors, but also to an infinite set of vectors in a vector space. | ||
Revision as of 01:36, 10 December 2011
Basis
Definition: The vectors v1, v2,..., vk in a vector space V are said to form a basis for V if (a) v1, v2,..., vk span V and (b) v1, v2,..., vk are linearly independent.
Note*
If v1, v2,..., vk form a basis for a vector space V, then they must be distinct and nonzero.
Note**
The above definition not only applies to a finite set of vectors, but also to an infinite set of vectors in a vector space.
