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.

*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 *a*_{1}*v*_{1}+ ... +*a*_{n}*v*_{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

**R**.

^{2}
Proof: We have to prove that these 2 vectors are both linearly independent and that they generate **R ^{2}**.

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.

**R**, we have to let (a,b) be an arbitrary element of

^{2}**R**, and show that there exist numbers x,y such that:

^{2}- 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 E

_{1}, E

_{2}, ..., E

_{n}are linearly independent and generate

**R**. Therefore, they form a basis for

^{n}**R**.

^{n}
**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 :*