zero map

Definition Suppose $X$ is a set, and $Y$ is a vector space with zero vector $0$. If $Z$ is a map $Z:X\to Y$, such that $Z(x)=0$ for all $x$ in $X$, then $Z$ is a zero map.

0.0.1 Examples

1. On the set of non-invertible $n\times n$ matrices, the determinant is a zero map.

2. If $X$ is the zero vector space, any linear map $T:X\to Y$ is a zero map. In fact, $T(0)=T(0\cdot 0)=0T(0)=0$.

3. If $X=Y$ and its field is $ℝ$ or $ℂ$, then the spectrum of $Z$ is $\{0\}$.

Title	zero map
Canonical name	ZeroMap
Date of creation	2013-03-22 14:03:38
Last modified on	2013-03-22 14:03:38
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	6
Author	matte (1858)
Entry type	Definition
Classification	msc 15-00
Related topic	ZeroVectorSpace
Related topic	ConstantFunction
Related topic	IdentityMap
Defines	zero operator