Three linearly independent vectors 
, 
and 
form a basis, because any vector in the space can be set as a linear combination of them.
The coordinates of the vector that form the base are:
Orthogonal Basis
The three basis vectors are mutually perpendicular.
Orthonormal Basis
The three basis vectors are mutually perpendicular and also have a length of one.
The base formed by 
, 
and 
is called the standard basis or canonical basis.
Examples
1. Demonstrate that the vectors = (1, 2, 3), = (2, 1, 0) and = (−1, −1, 0) form a basis and calculate the coordinates of the vector (1, −1, 0) on that basis.
The homogeneous system admits only the trivial solution:
Therefore, the three vectors are linearly independent and form a basis.
The coordinates of the vector (1, −1, 0) with respect to the basis are: 
.
1 Prove that the vectors (1, 1, 0), (1, 0, 1) and (0, 1, 1) form a basis.
The three vectors form a basis if they are linearly independent.
In the homogeneous system, the rank coincides with the number of unknowns, thus it admits only the trivial solution:
The vectors are linearly independent and therefore form a basis.
2Find the coordinates of the standard basis vectors on this basis.
3. Calculate the value of a for which the vectors 
, 
and 
form a basis.
If a ≠ 1, the vectors form a basis.
