Basis (linear algebra)

In mathematics, 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 includes a finiteness condition: a linear combination is always a finite sum of the form a₁v₁+ ... +a_nv_n.

Using Zorn's lemma, one can show that:

a) Every vector space has a basis.

b) Every basis of a vector space has the same cardinality, called the dimension of the vector space.. This result is known as the dimension theorem for vector spaces.

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.

A different notion is that of orthonormal basis of a Hilbert space and some other kinds of bases that occur in Banach spaces or, more generally, in topological vector spaces, where one may define inifinte sums (or series) and express elements of the space as infinite linear combinations of other elements. Ordinary bases are sometimes called Hamel bases in order to distiguish them from these topological concepts, .

See also :