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=='''Basis'''== | =='''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. | '''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. | |
| − | 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. |
| − | + | ||
| − | 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 02:36, 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.
