Delta-v

In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal and unit vectors. 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.

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,

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

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 length. 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.

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. $x_{1}x_{2}+y_{1}y_{2}=0\quad$
2. ${\sqrt {{x_{1}}^{2}+{y_{1}}^{2}}}=1$
3. ${\sqrt {{x_{2}}^{2}+{y_{2}}^{2}}}=1$

Converting from Cartesian to polar coordinates, and considering Equation $(2)$ and Equation $(3)$ immediately gives the result r₁ = r₂ = 1. In other words, requiring the vectors be of unit length restricts the vectors to lie on the unit circle.

After substitution, Equation $(1)$ becomes $\cos \theta _{1}\cos \theta _{2}+\sin \theta _{1}\sin \theta _{2}=0$. Rearranging gives $\tan \theta _{1}=-\cot \theta _{2}$. Using a trigonometric identity to convert the cotangent term gives

$\tan(\theta _{1})=\tan \left(\theta _{2}+{\tfrac {\pi }{2}}\right)$

$\Rightarrow \theta _{1}=\theta _{2}+{\tfrac {\pi }{2}}$

It is clear that in the plane, orthonormal vectors are simply radii of the unit circle whose difference in angles equals 90°.

Definition

Let ${\mathcal {V}}$ be an inner-product space. A set of vectors

$\left\{u_{1},u_{2},\ldots ,u_{n},\ldots \right\}\in {\mathcal {V}}$

is called orthonormal if and only if

$\forall i,j:\langle u_{i},u_{j}\rangle =\delta _{ij}$

where $\delta _{ij}\,$ is the Kronecker delta and $\langle \cdot ,\cdot \rangle$ is the inner product defined over ${\mathcal {V}}$.

Significance

Orthonormal sets are not especially significant on their own. However, they display certain features that make them fundamental in exploring the notion of diagonalizability of certain operators on vector spaces.

Properties

Orthonormal sets have certain very appealing properties, which make them particularly easy to work with.

• Theorem. If {e₁, e₂,...,e_n} is an orthonormal list of vectors, then

$||a_{1}e_{1}+a_{2}e_{2}+\cdots +a_{n}e_{n}||^{2}=|a_{1}|^{2}+|a_{2}|^{2}+\cdots +|a_{n}|^{2}$

• Theorem. Every orthonormal list of vectors is linearly independent.

Existence

• Gram-Schmidt theorem. If {v₁, v₂,...,v_n} is a linearly independent list of vectors in an inner-product space ${\mathcal {V}}$, then there exists an orthonormal list {e₁, e₂,...,e_n} of vectors in ${\mathcal {V}}$ such that span(e₁, e₂,...,e_n) = span(v₁, v₂,...,v_n).

Proof of the Gram-Schmidt theorem is constructive, and discussed at length elsewhere. The Gram-Schmidt theorem, together with the axiom of choice, guarantees that every vector space admits an orthonormal basis. This is possibly the most significant use of orthonormality, as this fact permits operators on inner-product spaces to be discussed in terms of their action on the space's orthonormal basis vectors. What results is a deep relationship between the diagonalizability of an operator and how it acts on the orthonormal basis vectors. This relationship is characterized by the Spectral Theorem.

Examples

Standard basis

The standard basis for the coordinate space Fⁿ is

{e1, e2,...,en}   where	e1 = (1, 0, ..., 0)
	e2 = (0, 1, ..., 0)
	$\vdots$
	en = (0, 0, ..., 1)

Any two vectors e_i, e_j where i≠j are orthogonal, and all vectors are clearly of unit length. So {e₁, e₂,...,e_n} forms an orthonormal basis.

Real-valued functions

When referring to real-valued functions, usually the L² inner product is assumed unless otherwise stated. Two functions $\phi (x)$ and $\psi (x)$ are orthonormal over the interval $[a,b]$ if

$(1)\quad \langle \phi (x),\psi (x)\rangle =\int _{a}^{b}\phi (x)\psi (x)dx=0,\quad {\rm {and}}$

$(2)\quad ||\phi (x)||_{2}=||\psi (x)||_{2}=\left[\int _{a}^{b}|\phi (x)|^{2}dx\right]^{\frac {1}{2}}=\left[\int _{a}^{b}|\psi (x)|^{2}dx\right]^{\frac {1}{2}}=1.$

Fourier series

The Fourier series is a method of expressing a periodic function in terms of sinusoidal basis functions. Taking C[−π,π] to be the space of all real-valued functions continuous on the interval [−π,π] and taking the inner product to be

$\langle f,g\rangle =\int _{-\pi }^{\pi }f(x)g(x)dx$

It can be shown that

$\left\{{\frac {1}{\sqrt {2\pi }}},{\frac {\sin(x)}{\sqrt {\pi }}},{\frac {\sin(2x)}{\sqrt {\pi }}},\ldots ,{\frac {\sin(nx)}{\sqrt {\pi }}},{\frac {\cos(x)}{\sqrt {\pi }}},{\frac {\cos(2x)}{\sqrt {\pi }}},\ldots ,{\frac {\cos(nx)}{\sqrt {\pi }}}\right\},\quad n\in \mathbb {N}$

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 in C[−π,π] and therefore an orthonormal basis of C[−π,π].

See also

• Orthogonalization

References

• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010