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ker

Represents the kernel (or null space) of a linear transformation in linear algebra and abstract algebra.

Overview

Essential in advanced mathematics for studying linear mappings and homomorphisms between algebraic structures.

  • Fundamental in linear algebra for understanding the set of all elements that map to zero
  • Critical in group theory and ring theory for analyzing homomorphisms
  • Commonly appears in theoretical computer science and cryptography
  • Used extensively in research papers and advanced mathematical texts when discussing linear transformations and abstract algebra

Examples

Finding the kernel of a linear transformation T.

ker⁡T={x∈V:T(x)=0}\ker T = \{x \in V : T(x) = 0\} 
\ker T = \{x \in V : T(x) = 0\}

Expressing the kernel of a matrix A.

ker⁡A={x∈Rn:Ax=0}\ker A = \{x \in \mathbb{R}^n : Ax = 0\} 
\ker A = \{x \in \mathbb{R}^n : Ax = 0\}

Relationship between kernel and image dimensions.

dim⁡ker⁡T+dim⁡im⁡T=dim⁡V\dim \ker T + \dim \operatorname{im} T = \dim V 
\dim \ker T + \dim \operatorname{im} T = \dim V