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




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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 4743 times total.

Classification:
AMS MSC15-00 (Linear and multilinear algebra; matrix theory :: General reference works )

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