Linear Algebra - Norm (Length)

> Linear Algebra

1 - About

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

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3 - Definition

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

where:

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).
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linear_algebra/norm.txt · Last modified: 2015/06/28 17:59 by gerardnico