ortonormaalinen

Ortonormaalinen is a Finnish term that translates to "orthonormal" in English. It is a concept used in linear algebra and functional analysis, particularly when dealing with vector spaces. A set of vectors is considered orthonormal if two conditions are met: orthogonality and normality.

Orthogonality means that each vector in the set is perpendicular to every other vector in the set. Mathematically, this is expressed by saying that the inner product (or dot product in Euclidean spaces) of any two distinct vectors in the set is zero. If the vectors are denoted by $v_i$ and $v_j$, then their inner product $<v_i, v_j>$ is zero when $i \neq j$.

Normality means that each vector in the set has a unit length, also known as a norm of one. The norm of a vector $v_i$ is typically denoted by $||v_i||$. For an orthonormal set, $||v_i|| = 1$ for all vectors $v_i$ in the set.

Combining these two conditions, a set of vectors $\{v_1, v_2, ..., v_n\}$ is orthonormal if $<v_i, v_j> = \delta_{ij}$, where $\delta_{ij}$ is the Kronecker delta. The Kronecker delta is equal to 1 if $i = j$ and 0 if $i \neq j$.

Orthonormal sets are extremely useful because they simplify many mathematical operations. For instance, expressing a vector as a linear combination of orthonormal basis vectors is straightforward. They also play a crucial role in areas like signal processing, quantum mechanics, and approximation theory, forming the basis for transformations like the Fourier transform and discrete cosine transform.