# Linear Algebra - Norm (Length)

### Table of Contents

## 1 - About

The norm of a vector v is written <math>\left \| v \right \|</math>

## 2 - Articles Related

## 3 - Definition

The norm of a vector v is defined by: <MATH>\left \| v \right \| = \sqrt{\left \langle v,v \right \rangle}</MATH>

where:

- <math>\langle v,v \rangle</math> is the inner product of v.

### 3.1 - Euclidean space

In Euclidean space, the inner product is the dot product.

<math> \begin{array}{crl} v & = &[v_1, v_2, \dots , v_n] \\ \left \| v \right \| & = & \left \| [v_1, v_2, . . . , v_n] \right \| \\ \left \| v \right \| & = & \sqrt{ {v_1}^2 + {v_2}^2 + \dots + {v_n}^2}\\ \left \| v \right \| & = & \sqrt{ \sum v^2_i }\\ \end{array} </math>

For a 2-vector:

<math>
\begin{array}{crl}
u & = & [u_1, u_2] \\
\left \| u \right \| & = & \sqrt{ {u_1}^2 + {u_2}^2 }\\
(\left \| u \right \|)^2 & = & {u_1}^2 + {u_2}^2 \\
\end{array}
</math>

as the Pythagorean theorem, the norm is then the geometric length of its arrow.

## 4 - Property

Since it plays the role of length, it should satisfy the following norm properties:

- Property N1: <math>\left \| v \right \|</math> is a non-negative real number.
- Property N2: <math>\left \| v \right \|</math> is zero if and only if v is a zero vector.
- Property N3: for any scalar <math>\alpha, \left \| \alpha.v \right \| = |\alpha | \left \| v \right \|</math>
- Property N4: <math>\left \| u + v \right \| = \left \| u \right \| + \left \| v \right \|</math> (triangle inequality).