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Basis of a vector space

A basis of a vector space is a set of linearly independent vectors that spans the whole space - every vector in the space can be written uniquely as a combination of the basis vectors.

basis: linearly independent vectors that span the space\text{basis: linearly independent vectors that span the space}definition of a basis

Symbols

dim⁡V\dim Vthe dimension of the space V, number of basis vectors

Example

In R3\mathbb R^3, e⃗1=(1,0,0), e⃗2=(0,1,0), e⃗3=(0,0,1)\vec e_1=(1,0,0),\ \vec e_2=(0,1,0),\ \vec e_3=(0,0,1) is the standard basis, and dim⁡R3=3\dim\mathbb R^3=3.

Every basis of the same vector space has the same number of vectors, equal to the dimension.
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