zero map

DefinitionSuppose $X$ 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 determinantis a zero map.

2. If $X$ is the zero vector space, any linear map $T:X\to Y$ isa 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$\{0\}$.