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 over the field "F", then :rank "A" + nullity "A" = "n". [harvtxt|Meyer|2000, page 199.]

This applies to linear maps as well. Let "V" and "W" be vector spaces over the field "F" and let "T" : "V" → "W" be a linear map. Then the rank of "T" is the dimension of the image of "T", the nullity the dimension of the kernel of "T", and we have: dim (im "T") + dim (ker "T") = dim "V"thus, equivalently,: rank "T" + nullity "T" = dim "V".This is in fact more general than the matrix statement above, because here "V" and "W" may even be infinite-dimensional.

One can refine this statement (via the splitting lemma or the below proof) to be a statement about an isomorphism of spaces, not just dimensions: in addition to::$dim\; operatorname\{im\},T\; +\; dim\; ker\; T\; =\; dim\; V$one in fact also has::$operatorname\{im\},T\; oplus\; ker\; T\; approx\; V.$

Proof

To prove the theorem, one starts with a basis of the kernel of "T", and extends it to a basis of all of "V". It is then not too difficult to show that "T" applied to the "new" basis vectors yields a basis of the image of "T".

In more abstract terms, the map $Tcolon\; V\; o\; operatorname\{image\},T$ "splits".

Reformulations and generalizations

This theorem is a statement of the first isomorphism theorem of algebra to the case of vector spaces; it generalizes to the splitting lemma.

In more modern language, the theorem can also be phrased as follows: if:0 → "U" → "V" → "R" → 0is a short exact sequence of vector spaces, then:dim("U") + dim("R") = dim("V")Here "R" plays the role of im "T" and "U" is ker "T".

In the finite-dimensional case, this formulation is susceptible to a generalization: if :0 → "V"₁ → "V"₂ → ... → "V"_"r" → 0is an exact sequence of finite-dimensional vector spaces, then:$sum\_\{i=1\}^r\; (-1)^idim(V\_i)\; =\; 0.$

The rank–nullity theorem for finite-dimensional vector spaces may also be formulated in terms of the "index" of a linear map. The index of a linear map "T" : "V" → "W", where "V" and "W" are finite-dimensional, is defined by:index "T" = dim(ker "T") - dim(coker "T"). Intuitively, dim(ker "T") is the number of independent solutions "x" of the equation "Tx" = 0, and dim(coker "T") is the number of independent restrictions that have to be put on "y" to make "Tx" = "y" solvable. The rank–nullity theorem for finite-dimensional vector spaces is equivalent to the statement:index "T" = dim("V") - dim("W").We see that we can easily read off the index of the linear map "T" from the involved spaces, without any need to analyze "T" in detail. This effect also occurs in a much deeper result: the Atiyah–Singer index theorem states that the index of certain differential operators can be read off the geometry of the involved spaces.

See also

* Splitting lemma

Notes

References

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