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ZixuanZhang
ZixuanZhang
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Image, kernel, rank and nullity

For a linear map , and are subspaces, with and .

Definition 3.2

Let be a linear map.

  • The image of under is the vector .

    The image of is the set

    It forms a subspace of .

  • If such that , then is in the kernel of .

    The kernel of is the set

    It forms a subspace of .

  • For , is called the domain of and the codomain of .

  • The dimension of the image of , , is called the rank of , denoted .

  • The dimension of the kernel of , , is called the nullity of , denoted .

Rank and nullity

The rank of is and the nullity is : the dimensions of the image and the kernel. Since is a subspace of and is a subspace of ,

For , is called the domain of and the codomain of . The image of a point under is the vector .

Examples

  • The zero linear map , defined by for all , has and .

  • The identity map , defined by , has and .

  • On , the map with

    is linear; here and .

Related

Stated in