What is Orthonormal Basis?

Introduction to Quantum Computing

The two vectors |0> and |1> are orthonormal means that they are both orthogonal and normalised.

Orthogonal means the vectors are at right angles to each other:

Normalized means the magnitudes of the vectors (i.e. length of the arrow) is equal to 1.

The two vectors |0> and |1> are linearly independent.  This means that they cannot be described  in terms of each other. However, using both the vectors |0> and |1>, and the rules of addition and multiplication of vectors by scalars, then all possible vectors in 2D space can be described:

Because the vectors |0> and |1> are linearly independent and can be used to describe any vector in 2D space by using vector addition and scalar multiplication, the vectors |0> and |1> form a basis. If the vectors are both orthogonal and normalized, they are called orthonormal basis.