Basis (linear algebra) : Basis of a vector space
A subset S of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions:- S is a minimal generating set of V.
- S is a maximal set of linearly independent vectors.
- S is both a set of linearly independent vectors and a generating set of V.
- every vector in V can be expressed as a linear combination of vectors in S in a unique way.
This definition tacitly includes a finiteness condition. A basis of a vector space is sometimes called a Hamel basis in order to distinguish it from the concept of an orthonormal basis of a Hilbert space and some other kinds of bases that occur in Banach spaces. An orthonormal basis of a Hilbert space H is an orthonormal set of members of H such that any member of the H can be written as a linear combination of a possibly infinite set of members of the orthonormal basis. In order to speak of infinite linear combinations, one must have a notion of convergence. Since Hilbert spaces are metric spaces, such a notion is available and natural.
Using Zorn's lemma, one can show that:
- every vector space has a basis.
Example I: Show that the vectors (1,1) and (-1,2) form a basis for R2.
Proof: We have to prove that these 2 vectors are both linearly independent and that they generate R2.
Part I: To prove that they are linearly independent, suppose that there are numbers a,b such that:
- a(1,1)+b(-1,2)=(0,0)
- (a-b,a+2b)=(0,0) and a-b=0 and a+2b=0.
- 3b=0 so b=0.
- a=0.
Part II: To prove that these two vectors generate R2, we have to let (a,b) be an arbitrary element of R2, and show that there exist numbers x,y such that:
- x(1,1)+y(-1,2)=(a,b)
- x-y=a
- x+2y=b.
- 3y=b-a, and then
- y=(b-a)/3, and finally
- x=y+a=((b-a)/3)+a.
Example II: We have already shown that E1, E2, ..., En are linearly independent and generate Rn. Therefore, they form a basis for Rn.
Example III: Let W be the vector space generated by et, e2t. We have already shown they are linearly independent. Then they form a basis for W.
- See also : Linear algebra
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