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Definitions from Kernel, image and rank

All definitions from Kernel, image and rank.

Back to Kernel, image and rank

Kernel

If T:V→WT:V\to W is a linear map, its kernel is

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

The same set is also called the null space. If A∈Fm×nA\in\mathbb F^{m\times n} is regarded as the map Fn→Fm\mathbb F^n\to\mathbb F^m, then

ker⁡(A)={x∈Fn:Ax=0}.\ker(A)=\{x\in\mathbb F^n:Ax=0\}.

The kernel lies in the domain because it records which inputs are sent to the zero vector. For the dot-product map Tu(x)=u⋅xT_u(x)=u\cdot x with u≠0u\ne0, the kernel is the subspace of vectors orthogonal to uu.

  1. [1]:Linear map
  2. [2]:Eigenspace
  3. [3]:Eigenvalues and eigenvectors
  4. [4]:Kernel of a matrix
  5. [5]:The kernel is a subspace
  6. 10 more

Nullity

The nullity of a linear map TT is

nullity⁡(T)=dim⁡ker⁡(T).\operatorname{nullity}(T)=\dim\ker(T).

For a matrix AA, the nullity is

nullity⁡(A)=dim⁡ker⁡(A).\operatorname{nullity}(A)=\dim\ker(A).

Nullity counts the independent input directions that are lost completely. For a matrix, it is the dimension of the solution subspace of Ax=0Ax=0: the number of free parameters after Gaussian elimination and back substitution, or the number of non-pivot columns in a row-echelon form. In the kernel-of-a-matrix example above, the single spanning vector is a basis for the kernel, so nullity⁡(A)=1\operatorname{nullity}(A)=1.

  1. [1]:Kernel
  2. [2]:Dimension
  3. [3]:Column independence and nullity
  4. [4]:Rank-nullity theorem
  5. [5]:Rank-nullity theorem for vector spaces
  6. [6]:A projection, nullity and a composition
  7. 1 more

Image

If T:V→WT:V\to W is a linear map, its image is

im⁡(T)={u∈W:u=T(v) for some v∈V}.\operatorname{im}(T)=\{u\in W:u=T(v)\text{ for some }v\in V\}.

The same set is also called the range. If A∈Fm×nA\in\mathbb F^{m\times n} is regarded as the map Fn→Fm\mathbb F^n\to\mathbb F^m, then

im⁡(A)={u∈Fm:u=Ax for some x∈Fn},\operatorname{im}(A)=\{u\in\mathbb F^m:u=Ax\text{ for some }x\in\mathbb F^n\},

which is the column space of AA.

The image lies in the codomain because it collects the outputs that can occur. For the dot-product map Tu(x)=u⋅xT_u(x)=u\cdot x with u≠0u\ne0, every real number occurs: for α∈R\alpha\in\mathbb R,

Tu(αu⋅uu)=α.T_u\left(\frac{\alpha}{u\cdot u}u\right)=\alpha.
  1. [1]:Linear map
  2. [2]:Column space
  3. [3]:Image of a matrix
  4. [4]:The image is a subspace
  5. [5]:Rank
  6. [6]:Properties read from a matrix representation
  7. 1 more

Rank

The rank of a linear map TT is

rank⁡(T)=dim⁡im⁡(T).\operatorname{rank}(T)=\dim\operatorname{im}(T).

For a matrix AA, the rank is

rank⁡(A)=dim⁡im⁡(A).\operatorname{rank}(A)=\dim\operatorname{im}(A).

For a matrix, rank is also the maximum number of linearly independent columns, or the number of pivot columns in a row-echelon form. In the image-of-a-matrix example above, the two displayed columns form a basis of the image, so rank⁡(A)=2\operatorname{rank}(A)=2.

“Pivot columns” refers to positions in the echelon form. When row operations have been used, take the columns in those positions from the original matrix to obtain a basis for the image.

  1. [1]:Image
  2. [2]:Dimension
  3. [3]:Rank-nullity theorem
  4. [4]:Rank-nullity theorem for vector spaces
  5. [5]:Kernels, ranks and powers of a matrix
  6. [6]:Ranks of products involving a matrix and its transpose
  7. 1 more

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