Orthogonal

EXPLANATION Formally, two Vectors $x$ and $y$ in an Inner Product Space $V$ are orthogonal if their inner product $\langle x, y angle$ is zero. This situation is denoted $x \perp y$ . Two Vector Subspaces $A$ and $B$ of Vector Space $V$ are called orthogonal subspaces if each vector in $A$ is orthogonal to each vector in $B$ . The largest subspace that is orthogonal to a given subspace is its Orthogonal Complement . A . That is, for all pairs of vectors $x$ and $y$ in the inner product space $V$ , : $\langle Tx, Ty angle = \langle x, y angle.$ This means that $T$ preserves the Angle between $x$ and $y$ , and that the Lengths of $Tx$ and $x$ are equal. A Term Rewriting System is said to be Orthogonal if it is left-linear and is non-ambiguous. Orthogonal term rewriting systems are Confluent . The word normal is sometimes also used in place of orthogonal. However, ''normal'' can also refer to Unit Vectors . In particular, Orthonormal refers to a collection of vectors that are both orthogonal and normal (of unit length). So, using the term ''normal'' to mean "orthogonal" is often avoided. In some contexts, two things are said to be orthogonal if they are mutually exclusive. IN EUCLIDEAN VECTOR SPACES In 2- or 3- Dimension al Euclidean Space , two vectors are orthogonal if their Dot Product is zero, i.e. they make an angle of 90° or π/2 Radian s. Hence orthogonality of vectors is a generalization of the concept of Perpendicular . In terms of Euclidean Subspace s, the orthogonal complement of a Line is the Plane perpendicular to it, and vice versa. Note however that there is no correspondence with regards to perpendicular planes, because vectors in subspaces start from the Origin . In 4-dimensional Euclidean space, the orthogonal complement of a line is a Hyperplane and vice versa, and that of a plane is a plane. Several vectors are called pairwise orthogonal if any two of them are orthogonal, and a set of such vectors is called an '''orthogonal set'''. Such a set is an '''orthonormal set''' if all its vectors are Unit Vector s. Non-zero pairwise orthogonal vectors are always Linearly Independent . ORTHOGONAL FUNCTIONS It is common to use the following inner product for two Function s ''f'' and ''g'': : $\langle f, g angle_w = \int_a^b f(x)g(x)w(x)\,dx.$ Here we introduce a nonnegative Weight Function $w(x)$ in the definition of this inner product. We say that those functions are orthogonal if that inner product is zero: : $\int_a^b f(x)g(x)w(x)\,dx = 0.$ We write the Norm s with respect to this inner product and the weight function as
	:<math>\langle F I, F J Angle	\int_{-\infty}^\infty f_i(x) f_j(x) w(x)\,dx=f_i^2\delta_{i,j}=f_j^2\delta_{i,j}</math>

:for some positive integer ''a'', and for 1 ≤ ''k'' ≤ ''a'' − 1, these vectors are orthogonal, for example (1, 0, 0, 1, 0, 0, 1, 0)^T, (0, 1, 0, 0, 1, 0, 0, 1)^T, (0, 0, 1, 0, 0, 1, 0, 0)^T are orthogonal.

Take two quadratic functions 2''t'' + 3 and 5''t''² + ''t'' − 17/9. These functions are orthogonal with respect to a unit weight function on the interval from −1 to 1. The product of these two functions is 10''t''³ + 17''t''² − 7/9 ''t'' − 17/3, and now,

\int_{-1}^{1} \left(10t^3+17t^2-{7\over 9}t-{17\over 3}
ight)\,dt = \left 2}t^4+{17\over 3}t^3-{7\over 18}t^2-{17\over 3}t
ight_{-1}^{1}

=\left({5\over 2}(1)^4+{17\over 3}(1)^3-{7\over 18}(1)^2-{17\over 3}(1)
ight)-\left({5\over 2}(-1)^4+{17\over 3}(-1)^3-{7\over 18}(-1)^2-{17\over 3}(-1)
ight)

={19\over 9}-{19\over 9}=0.

The functions 1, sin(''nx''), cos(''nx'') : ''n'' = 1, 2, 3, ... are orthogonal with respect to Lebesgue Measure on the interval from 0 to 2π. This fact is basic in the theory of Fourier Series .

Various eponymously named polynomial sequences are sequences of Orthogonal Polynomials . In particular:

---The Hermite Polynomials are orthogonal with respect to the Normal Distribution with expected value 0.

---The Legendre Polynomials are orthogonal with respect to the Uniform Distribution on the interval from −1 to 1.

---The Laguerre Polynomials are orthogonal with respect to the Exponential Distribution . Somewhat more general Laguerre polynomial sequences are orthogonal with respect to Gamma Distribution s.

---The Chebyshev Polynomials of the first kind are orthogonal with respect to the measure $1/\sqrt{1-x^2}.$

---The Chebyshev polynomials of the second kind are orthogonal with respect to the Wigner Semicircle Distribution .

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Orthogonal