properties of spanning sets
If , then for . But by assumption. So as well. If , and , then . ∎
If contains , then .
Let . So by 1 above. If , then . If one of the ’s, say , is , then . ∎
If is a basis for , then spans and is linearly independent. Let be the set obtained from with deleted. If spans , then can be written as a linear combination of elements in . But then would no longer be linearly independent, contradiction the assumption. Therefore, is minimal.
Conversely, suppose is a minimal spanning set for . Furthermore, suppose that is linearly dependent. Let , with . Then
where . So any linear combination of elements in involving can be replaced by a linear combination not involving through equation (1). Therefore . But this means that is not minimal, contrary to our assumption. Therefore, must be linearly independent. ∎
Remark. All of the properties above can be generalized to modules over rings, except the last one, where the implication is only one-sided: basis implying minimal spanning set.
|Title||properties of spanning sets|
|Date of creation||2013-03-22 18:05:40|
|Last modified on||2013-03-22 18:05:40|
|Last modified by||CWoo (3771)|