Linear map

Kernel of a linear map

The kernel ker⁡𝑇 or null space of a linear map 𝑇 ∈𝖵𝖾𝖼𝗍𝕂(𝑈,𝑉) is the the preïmage 𝑇−1{⃗𝟎}, #m/def/linalg i.e. the set of all vectors in 𝑈 that are mapped to ⃗𝟎. It is therefore equivalent to the Kernel of a group homomorphism of 𝑇 considered as a group homomorphism. The nullity nullity⁡𝑇 of a linear map is the dimension of its kernel. #m/def/linalg

Properties

• Rank-nullity theorem
• If ⃗𝐮 is a solution to 𝐴⃗𝐯 =⃗𝟎 then the full solution set is ⃗𝐮 +ker⁡𝐴

#state/tidy | #lang/en | #SemBr