Linear Algebra - Norm (Length)

The norm of a vector v is written $\left \| v \right \|$

3 - Definition

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

where:

3.1 - Euclidean space

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

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

For a 2-vector:
$\begin{array}{crl} u & = & [u_1, u_2] \\ \left \| u \right \| & = & \sqrt{ {u_1}^2 + {u_2}^2 }\\ (\left \| u \right \|)^2 & = & {u_1}^2 + {u_2}^2 \\ \end{array}$
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: $\left \| v \right \|$ is a non-negative real number.
• Property N2: $\left \| v \right \|$ is zero if and only if v is a zero vector.
• Property N3: for any scalar $\alpha, \left \| \alpha.v \right \| = |\alpha | \left \| v \right \|$
• Property N4: $\left \| u + v \right \| = \left \| u \right \| + \left \| v \right \|$ (triangle inequality).