Rank–nullity theorem
In mathematics, the rank–nullity theorem of linear algebra, in its simplest form, states that the rank and the nullity of a matrix add up to the number of columns of the matrix. Specifically, if A is an m-by-n matrix (with m rows and n columns) over some field, then This applies to linear maps as well. Let V and W be vector spaces over some field and let T : V → W be a linear map. Then the rank of T is the dimension of the image of T and the nullity of T is the dimension of the kernel of T, so we have or, equivalently,
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Rank–nullity theorem
In mathematics, the rank–nullity theorem of linear algebra, in its simplest form, states that the rank and the nullity of a matrix add up to the number of columns of the matrix. Specifically, if A is an m-by-n matrix (with m rows and n columns) over some field, then This applies to linear maps as well. Let V and W be vector spaces over some field and let T : V → W be a linear map. Then the rank of T is the dimension of the image of T and the nullity of T is the dimension of the kernel of T, so we have or, equivalently,
has abstract
Der Rangsatz oder Dimensionssa ...... wischen zwei Vektorräumen auf.
@de
Em matemática, mais especifica ...... fundamental da álgebra linear.
@pt
En matemáticas, el teorema ran ...... ue tenemos o equivalentemente,
@es
En mathématiques, et plus préc ...... ou d'une matrice par son rang.
@fr
In algebra lineare, il teorema ...... mero di colonne della matrice.
@it
In mathematics, the rank–nulli ...... tal theorem of linear algebra.
@en
Twierdzenie o rzędzie – twierd ...... erdzenie Kroneckera-Capellego.
@pl
数学の線型代数学の分野における階数・退化次数の定理（かいすう ...... 関連付けられる像、核、余像、余核について考えることが出来る。
@ja
秩－零化度定理是线性代数中的一个定理，给出了一个线性变换或一 ...... 实际上定理在更广的范围内也成立，因为 和 可以是无限维的。
@zh
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Der Rangsatz oder Dimensionssa ...... wischen zwei Vektorräumen auf.
@de
Em matemática, mais especifica ...... o tem-se ou, equivalentemente,
@pt
En matemáticas, el teorema ran ...... de T (ker T), así que tenemos
@es
En mathématiques, et plus préc ...... ou d'une matrice par son rang.
@fr
In algebra lineare, il teorema ...... mero di colonne della matrice.
@it
In mathematics, the rank–nulli ...... , so we have or, equivalently,
@en
Twierdzenie o rzędzie – twierd ...... na jej obraz rozszczepia się.
@pl
数学の線型代数学の分野における階数・退化次数の定理（かいすう ...... 関連付けられる像、核、余像、余核について考えることが出来る。
@ja
秩－零化度定理是线性代数中的一个定理，给出了一个线性变换或一 ...... 实际上定理在更广的范围内也成立，因为 和 可以是无限维的。
@zh
label
Rangsatz
@de
Rank–nullity theorem
@en
Teorema del rango
@it
Teorema do núcleo e da imagem
@pt
Teorema rango-nulidad
@es
Théorème du rang
@fr
Twierdzenie o rzędzie
@pl
秩－零化度定理
@zh
階数・退化次数の定理
@ja
