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Basis (linear algebra)

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.

Recall that a set S is a generating set of V if every vector in V is a linear combination of vectors in S.

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.

Every basis of a vector space has the same cardinality, called the dimension of the vector space.

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)

Then:

(a-b,a+2b)=(0,0) and a-b=0 and a+2b=0.

Subtracting the first equation from the second, we obtain:

3b=0 so b=0.

And from the first equation then:

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)

Then we have to solve the equations:

x-y=a
x+2y=b.

Subtracting the first equation from the second, we get:

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, ..., E_n are linearly independent and generate Rⁿ. Therefore, they form a basis for Rⁿ.

Example III: Let W be the vector space generated by e^t, e^2t. We have already shown they are linearly independent. Then they form a basis for W.

See also : Linear algebra

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