# Lineaer algebra - orthogonality

• yy205001
That is not a complete definition of an orthonormal basis. An orthonormal basis is a basis where all the basis vectors are orthogonal to each other (inner product is 0) and each vector has length 1. So <e1, e1> = 1, <e2, e2> = 1, etc.
yy205001

## Homework Statement

Let<x,y> be an inner product on a vector space V, and let e1, e2,...,en be an orthonormal basis for V.
Prove: <x,y> = <x,e1><y,e1>+...+<x,en><y,en>.

## Homework Equations

<x,x> = abs(x)
a<x,y> = <ax,y> = <x,ay>

## The Attempt at a Solution

RHS
=<x,y>
= x1y1+...+xnyn
= x1<e1,e1>y1<e1,e1>+...+xn<en,en>yn<en,en>
* <e,e>=1
=e12<x1,e1>+...+en2<xn,en>

And i cannot find a way to take the e2 out of the equation.
Any help is appreciated

How is an orthonormal basis defined?

A subset {v_1,...,v_k} of a vector space V, with the inner product <,>, is called orthonormal if <v_i,v_j>=0 when i≠j. That is, the vectors are mutually perpendicular. Moreover, they are all required to have length one: <v_i,v_i>=1.

ehild

Orthonormal basis
A set of vectors for an inner product space whose are linearly independence and span V and orthonormal.

orthonormal: orthogonal (<x,y>=0) and each vector has length one.

Under "relevant equations" you have "<x, x>= abs(x)". There is NO "x2" so what, exactly do you mean by "en2"? <en, en>?
You also say "orthonormal: orthogonal (<x,y>=0) and each vector has length one."

## 1. What is orthogonality in linear algebra?

Orthogonality in linear algebra refers to the relationship between two vectors that are perpendicular to each other. This means that their dot product is equal to 0, indicating that they are at a 90-degree angle to each other.

## 2. How is orthogonality useful in linear algebra?

Orthogonality is useful in linear algebra because it allows for simplification of calculations and helps in finding solutions to systems of equations. It also aids in understanding geometric transformations and projections.

## 3. What is an orthogonal basis?

An orthogonal basis is a set of vectors that are mutually orthogonal, meaning that each pair of vectors in the set is orthogonal to each other. These basis vectors are linearly independent and can be used to represent any vector in the vector space.

## 4. How is orthogonality related to the concept of linear independence?

Orthogonality is closely related to the concept of linear independence. In fact, a set of vectors is linearly independent if and only if they are mutually orthogonal. This means that none of the vectors in the set can be expressed as a linear combination of the others.

## 5. Can orthogonal vectors be in more than two dimensions?

Yes, orthogonal vectors can exist in any number of dimensions. In three-dimensional space, for example, three vectors can be mutually orthogonal. In higher dimensions, more than three orthogonal vectors can exist, but it becomes harder to visualize.

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