properties of linear independence
is never linearly independent if .
Since . ∎
If is linearly independent, so is any subset of . As a result, if and are linearly independent, so is . In addition, is linearly independent, its spanning set being the singleton consisting of the zero vector .
If , where , then , so for all . ∎
If is a chain of linearly independent subsets of , so is their union.
Let be the union. If , then , for each . Pick the largest so that all ’s are in it. Since this set is linearly independent, for all . ∎
is a basis for iff is a maximal linear independent subset of . Here, maximal means that any proper superset of is linearly dependent.
If is a basis for , then it is linearly independent and spans . If we take any vector , then can be expressed as a linear combination of elements in , so that is no longer linearly independent, for the coefficient in front of is non-zero. Therefore, is maximal.
Conversely, suppose is a maximal linearly independent set in . Let be the span of . If , pick an element . Suppose , where , then . If , then would be in the span of , contradicting the assumption. So , and as a result, , since is linearly independent. This shows that is linearly independent, which is impossible since is assumed to be maximal. Therefore, . ∎
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 maximal linear independence.
|Title||properties of linear independence|
|Date of creation||2013-03-22 18:05:37|
|Last modified on||2013-03-22 18:05:37|
|Last modified by||CWoo (3771)|