0 denotes the zero vector in W. Kernel and image of linear algebra and its applications by gilbert strang pdf download map L. The kernel of L is a linear subspace of the domain V.
This is the generalization to linear operators of the row space, or coimage, of a matrix. The notion of kernel applies to the homomorphisms of modules, the latter being a generalization of the vector space over a field to that over a ring. The domain of the mapping is a module, and the kernel constitutes a “submodule”.
Here, the concepts of rank and nullity do not necessarily apply. W is continuous if and only if the kernel of L is a closed subspace of V. 0 is understood as the zero vector.
The dimension of the kernel of A is called the nullity of A. Thus the kernel of A is the same as the solution set to the above homogeneous equations. A over a field K is a linear subspace of Kn. This follows from the distributivity of matrix multiplication over addition.
The row space, or coimage, of a matrix A is the span of the row vectors of A. By the above reasoning, the kernel of A is the orthogonal complement to the row space. That is, a vector x lies in the kernel of A if and only if it is perpendicular to every vector in the row space of A. The dimension of the row space of A is called the rank of A, and the dimension of the kernel of A is called the nullity of A.
0T, where T denotes the transpose of a column vector. The left null space of A is the same as the kernel of AT.
The left null space of A is the orthogonal complement to the column space of A, and is dual to the cokernel of the associated linear transformation. The kernel, the row space, the column space, and the left null space of A are the four fundamental subspaces associated to the matrix A. A by the vector v.