Basis Vector

Three linearly independent vectors Vector U, Vector V and Vector W form a basis, because any vector in the space can be set as a linear combination of them.

Forming a Basis

The coordinates of the vector that form the base are:

Coordinates

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.

Orthonormal Basis

Basis Vectors

Orthonormal Basis

Perpendicular Vectors

The base formed by Vector I, Vector J and Vector K is called the standard basis or canonical basis.

Examples

1. Demonstrate that the vectors Vector U = (1, 2, 3), Vector V = (2, 1, 0) and Vector W = (−1, −1, 0) form a basis and calculate the coordinates of the vector (1, −1, 0) on that basis.

Linear Combination

Linear Combination

Linear Combination

System of Equations

The homogeneous system admits only the trivial solution:

Homogeneous System

Therefore, the three vectors are linearly independent and form a basis.

Linearly Independent

Linearly Independent

System of Equations

The coordinates of the vector (1, −1, 0) with respect to the basis are: Vector.


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.

Linearly Independent

Linearly Independent

System of Equations

In the homogeneous system, the rank coincides with the number of unknowns, thus it admits only the trivial solution:

Homogeneous System

The vectors are linearly independent and therefore form a basis.

2Find the coordinates of the standard basis vectors on this basis.

Linear Combination

Linear Combination

System of Equations

Systems

Coordinates

Linear Combination

Coordinates

Linear Combination

Coordinates


3. Calculate the value of a for which the vectors Vector U, Vector V and Vector W form a basis.

Solution

If a ≠ 1, the vectors form a basis.