Orthonormality

 ( Linear Algebra ) 

In linear algebra, two vector space in an inner product space are orthonormal if they are orthogonality . A unit vector means that the vector has a length of 1, which is also known as normalized. Orthogonal means that the vectors are all perpendicular to each other. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of unit length. An orthonormal set which forms a basis is called an orthonormal basis.

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Intuitive overview The construction of orthogonality of vectors is motivated by a desire to extend the intuitive notion of perpendicular vectors to higher-dimensional spaces. In the Cartesian plane, two vectors are said to be perpendicular if the angle between them is 90° (i.e. if they form a right angle). This definition can be formalized in Cartesian space by defining the dot product and specifying that two vectors in the plane are orthogonal if their dot product is zero.

Similarly, the construction of the norm of a vector is motivated by a desire to extend the intuitive notion of the length of a vector to higher-dimensional spaces. In Cartesian space, the norm of a vector is the square root of the vector dotted with itself. That is,

$\backslash |\; \backslash mathbf\{x\}\; \backslash |\; =\; \backslash sqrt\{\; \backslash mathbf\{x\}\; \backslash cdot\; \backslash mathbf\{x\}\}$

Many important results in linear algebra deal with collections of two or more orthogonal vectors. But often, it is easier to deal with vectors of Unit vector. That is, it often simplifies things to only consider vectors whose norm equals 1. The notion of restricting orthogonal pairs of vectors to only those of unit length is important enough to be given a special name. Two vectors which are orthogonal and of length 1 are said to be orthonormal.

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Simple example What does a pair of orthonormal vectors in 2-D Euclidean space look like?

Let u = (x₁, y₁) and v = (x₂, y₂). Consider the restrictions on x₁, x₂, y₁, y₂ required to make u and v form an orthonormal pair.

• From the orthogonality restriction, u • v = 0.
• From the unit length restriction on u, || u|| = 1.
• From the unit length restriction on v, || v|| = 1.

Expanding these terms gives 3 equations:

1. $x\_1\; x\_2\; +\; y\_1\; y\_2\; =\; 0\; \backslash quad$
2. $\backslash sqrt,\; \backslash frac\{\backslash sin(2x)\}\{\backslash sqrt\{\backslash pi\}\},\; \backslash ldots,\; \backslash frac\{\backslash sin(nx)\}\{\backslash sqrt\{\backslash pi\}\},\; \backslash frac\{\backslash cos(x)\}\{\backslash sqrt\{\backslash pi\}\},\; \backslash frac\{\backslash cos(2x)\}\{\backslash sqrt\{\backslash pi\}\},\; \backslash ldots,\; \backslash frac\{\backslash cos(nx)\}\{\backslash sqrt\{\backslash pi\}\}\; \backslash right\backslash \},\; \backslash quad\; n\; \backslash in\; \backslash mathbb\{N\}$

forms an orthonormal set.

However, this is of little consequence, because C−π,π is infinite-dimensional, and a finite set of vectors cannot span it. But, removing the restriction that n be finite makes the set Dense subset in C−π,π and therefore an orthonormal basis of C−π,π.

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See also

• Orthogonalization
• Orthonormal function system

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Sources

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