## Subsection1.2.2What is a vector norm?

A vector norm extends the notion of an absolute value to vectors. It allows us to measure the magnitude (or length) of a vector. In different situations, a different measure may be more appropriate.

###### Definition1.2.2.1. Vector norm.

Let $\nu: \Cm \rightarrow \mathbb R \text{.}$ Then $\nu$ is a (vector) norm if for all $x, y \in \Cm$ and all $\alpha \in \mathbb C$

• $x \neq 0 \Rightarrow \nu( x ) > 0$ ($\nu$ is positive definite),
• $\nu( \alpha x ) = \vert \alpha \vert \nu( x )$ ($\nu$ is homogeneous), and
• $\nu(x + y ) \leq \nu( x ) + \nu( y )$ ($\nu$ obeys the triangle inequality).

###### Homework1.2.2.1.

TRUE/FALSE: If $\nu: \Cm \rightarrow \mathbb R$ is a norm, then $\nu( 0 ) = 0 \text{.}$

Hint

From context, you should be able to tell which of these $0$'s denotes the zero vector of a given size and which is the scalar $0\text{.}$

$0 x = 0$ (multiplying any vector $x$ by the scalar $0$ results in a vector of zeroes).

Let $x \in \Cm$ and, just for clarity this first time, $\vec{0}$ be the zero vector of size $m$ so that $0$ is the scalar zero. Then
We typically use $\| \cdot \|$ instead of $\nu( \cdot )$ for a function that is a norm.