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zero map (Definition)

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.

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 $ \mathbb{R}$ or $ \mathbb{C}$, then the spectrum of $ Z$ is $ \{0\}$.



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"zero map" is owned by matte. [ full author list (2) | owner history (1) ]
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See Also: zero vector space, constant function, identity map

Also defines:  zero operator
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Cross-references: spectrum, field, linear map, zero vector space, determinant, matrices, non-invertible, map, zero vector, vector space
There are 11 references to this entry.

This is version 3 of zero map, born on 2003-11-01, modified 2005-12-11.
Object id is 5416, canonical name is ZeroMap.
Accessed 3629 times total.

Classification:
AMS MSC: 15-00 (Linear and multilinear algebra; matrix theory :: General reference works )
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