Basis Vector

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.