zero map (original) (raw)

DefinitionSuppose X is a set, and Y is a vector space with zero vector 0. If Z is a map Z:X→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×n matrices, the determinantis a zero map.
2. 1. If X is the zero vector space, any linear map T:X→Y is a zero map. In fact, T⁢(0)=T⁢(0⋅0)=0⁢T⁢(0)=0.
3. 1. If X=Y and its field is ℝ or ℂ, then the spectrum of Z is{0}.