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. 1.

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

2. 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. 3.

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

Title zero map ZeroMap 2013-03-22 14:03:38 2013-03-22 14:03:38 matte (1858) matte (1858) 6 matte (1858) Definition msc 15-00 ZeroVectorSpace ConstantFunction IdentityMap zero operator