Website owner:  James Miller

        ELEMENTARY BASES FOR VARIOUS LINEAR SPACES 

1. Space of all n-vectors of Euclidean n-dimensional space over some field F (real field, complex field, etc.) The following set of n elementary vectors constitute a basis:

                           E₁ = (1,0, ..., 0)^T

                           E₂ = (0,1, ..., 0)^T

                            ........................

                           E_n = (0,0, ..., 1)^T

2. Space of all mxn matrices over some field F (real field, complex field, etc.) The following set of mn mxn matrices constitute a basis:

                        { I = 1,m; j = 1,n }

            where has a 1 in the i-th row and j-th column, all other entries being zero

Example. A basis for the linear space of all 2x3 matrices is the set of six 2x3 matrices:

3. Space of all polynomials a₀ + a₁x + a₂x² + ... + a_nxⁿ of degree n. The set of n+1 vectors 1,x,x², ... ,xⁿ constitute a basis.