Kernel (linear algebra)

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the kernel of a <a href="/facts/Linear_map/Q1PBKNVV">linear map</a>, also known as the null space or nullspace, is the part of the <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a> which is mapped to the <a href="/facts/Zero_element/PvH2qhxh">zero vector</a> of the <a href="/facts/Codomain/MGc43nwQ">co-domain</a>; the kernel is always a <a href="/facts/Linear_subspace/2wtKTgHL">linear subspace</a> of the domain. That is, given a linear map L : V → W between two <a href="/facts/Vector_space/M1pxMLx2">vector spaces</a> V and W, the kernel of L is the vector space of all elements v of V such that L(v) = 0, where 0 denotes the <a href="/facts/Zero_vector/PvH2qhxh">zero vector</a> in W, or more symbolically:

ker
        ⁡
        (
        L
        )
        =
        
          {
          
            
              v
            
            ∈
            V
            ∣
            L
            (
            
              v
            
            )
            =
            
              0
            
          
          }
        
        =
        
          L
          
            −
            1
          
        
        (
        
          0
        
        )
        .
      
    
    {\displaystyle \ker(L)=\left\{\mathbf {v} \in V\mid L(\mathbf {v} )=\mathbf {0} \right\}=L^{-1}(\mathbf {0} ).}

Kernel (linear algebra) open-in-new

Kernel (linear algebra)