An orthonormal basis is a set of vectors in a vector space that are mutually orthogonal (perpendicular) and have a unit length. This means that all the vectors are independent of each other and have a magnitude of 1.
To prove that a basis is orthonormal, you need to show that the vectors are orthogonal (their dot product is equal to 0) and that each vector has a magnitude of 1. This can be done using the Gram-Schmidt process or by directly calculating the dot products and magnitudes of the vectors.
The inner product in linear algebra is a mathematical operation that takes two vectors and produces a scalar (number) as the result. It is also known as the dot product and is calculated by multiplying the corresponding components of the two vectors and then adding them together.
The uniqueness of an inner product can be proven by showing that it satisfies the four axioms of an inner product: linearity, positive definiteness, conjugate symmetry, and additivity. If these axioms are satisfied, then the inner product is unique.
Proving the orthonormality of a basis and the uniqueness of an inner product ensures that the vectors being used are independent of each other and that the inner product is well-defined. This is crucial in many applications of linear algebra, such as solving systems of equations and finding eigenvalues and eigenvectors.