Algebra
Presenter:Yanlu Dong
VCNS lab
Mechanical Engineering Department,Ajou University
Section Bases and Dimension
Content
Ordered and unordered set
Span
Linearly independent
Basis
dimension
Ordered and unordered set
Ordered and unordered set
Ordered set: (a,b) (a,b) is different from (b,a)
(a,a,b)is different from (a,b)
Unordered set:{a,b} {a,b} is same with{b,a}
{a,a,b} is same with {a,b}
Let V be a vector space over a field F, let be an ordered set of elements of V. A bination of is any vector of the form:
Span
The space spanned by a set S will often be denoted by Span S.
Span is the smallest subspace of V which contains S.
The order of the set is irrelevant here, the span of S is the same as the span of any reordering of S.
Proposition:
let S be a set of V, and W be a subspace of V. If S W,
then span S W.
Linearly independent
A linear relation among vectors is any relation of the form
where the coefficients are in F.
An ordered set of vectors is called linearly independent if there is no linear relation among the vectors in the set except all the are zero.
It is useful to state this condition positively:
A set which is not linearly independent is called linearly dependent
Basis
A set of vectors which is linearly independent and which also spans V is called a basis.
Let is a basis. Then since B spans V, every w V can be written as a bination. Since B is linearly independent, this expression is unique.
Proposition:
Let set is a basis if and only if every vector w V can be written in a unique way.
on the other hand, the definition of linear independence for B can be restated by saying that 0 has only one expression as a bination.
Basis
Proposition:
Let L be a linearly independent ordered set in V, and let v V be any vector. Then the ordered set obtained by adding v to L is linearly independent if and only if v is not in the subspace spanned by L.
Proof
say that . If v Span L, then for some . Hence ,
is a l
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