Rank Nullity Theorem of Linear Algebra

Transcription

1 Rank Nullity Theorem of Linear Algebra Jose Divasón and Jesús Aransay March 12, 2013 Abstract In this article we present a proof of the result known in Linear Algebra as the rank nullity Theorem, which states that, given any linear form f from a finite dimensional vector space V to a vector space W, then the dimension of V is equal to the dimension of the kernel of f (which is a subspace of V ) and the dimension of the range of f (which is a subspace of W ). The proof presented here is based on the one given in [1]. It makes use of the HOL-Multivariate-Analysis session of Isabelle, and of several of its results and definitions. As a corollary of the previous theorem, and taking advantage of the relationship between linear forms and matrices, we prove that, for every matrix A (which has associated a linear form between finite dimensional vector spaces), the sum of its null space and its column space (which is equal to the range of the linear form) is equal to the number of columns of A. Contents 1 Rank Nullity Theorem of Linear Algebra Previous results The proof The rank nullity theorem for matrices Rank Nullity Theorem of Linear Algebra theory Dim-Formula imports /src/hol/multivariate-analysis/multivariate-analysis begin This research has been funded by the project MTM C02-01 of the Spanish Government, the FET project of the 7th Framework program of the EU FORMATH (n ) and the research grant FPIUR12 of the Universidad de La Rioja. 1

2 1.1 Previous results Linear dependency is a monotone property, based on the monotonocity of linear independence: lemma dependent-mono: assumes d:dependent A and A-in-B: A B shows dependent B The negation of dependent P = ( S u. finite S S P ( v S. u v 0 ( v S. u v R v) = (0 :: a))) produces the following result: lemma independent-explicit: independent A = ( S A. finite S ( u. ( v S. u v R v) = 0 ( v S. u v = 0 ))) A finite set A for which every of its linear combinations equal to zero requires every coefficient being zero, is independent: lemma independent-if-scalars-zero: assumes fin-a: finite A and sum: f. ( x A. f x R x) = 0 ( x A. f x = 0 ) shows independent A Given a finite independent set, a linear combination of its elements equal to zero is possible only if every coefficient is zero: lemma scalars-zero-if-independent: assumes fin-a: finite A and ind: independent A and sum: ( x A. f x R x) = 0 shows x A. f x = 0 In an euclidean space, every set is finite, and thus [finite A; independent A; ( x A. f x R x) = (0 :: a)] = x A. f x = 0 holds: corollary scalars-zero-if-independent-euclidean: fixes A:: a::euclidean-space set assumes ind: independent A and sum: ( x A. f x R x) = 0 shows x A. f x = 0 2

3 The following lemma states that every linear form is injective over the elements which define the basis of the range of the linear form. This property is applied later over the elements of an arbitrary basis which are not in the basis of the nullifier or kernel set (i.e., the candidates to be the basis of the range space of the linear form). Thanks to this result, it can be concluded that the cardinal of the elements of a basis which do not belong to the kernel of a linear form f is equal to the cardinal of the set obtained when applying f to such elements. The application of this lemma is not usually found in the pencil and paper proofs of the Rank nullity theorem, but will be crucial to know that, being f a linear form from a finite dimensional vector space V to a vector space V, and given a basis B of ker f, when B is completed up to a basis of V with a set W, the cardinal of this set is equal to the cardinal of its range set: lemma inj-on-extended: and f : finite C and ind-c : independent C and C-eq: C = B W and disj-set: B W = {} and span-b: {x. f x = 0 } span B shows inj-on f W The proof is carried out by reductio ad absurdum 1.2 The proof Now the rank nullity theorem can be proved; given any linear form f, the sum of the dimensions of its kernel and range subspaces is equal to the dimension of the source vector space. It is relevant to note that the source vector space must be finite-dimensional (this restriction is introduced by means of the euclidean space type class), whereas the destination vector space may be finite or infinite dimensional (and thus a real vector space is used); this is the usual way the theorem is stated in the literature. The statement of the rank nullity theorem for linear algebra, as well as its proof, follow the ones on [1]. The proof is the traditional one found in the literature. The theorem is also named fundamental theorem of linear algebra in some texts (for instance, in [2]). theorem rank-nullity-theorem: assumes l: linear (f ::( a::{euclidean-space}) => ( b::{real-vector})) shows DIM ( a::{euclidean-space}) = dim {x. f x = 0 } + dim (range f ) 3

4 1.3 The rank nullity theorem for matrices The previous lemma can be moved to the matrices representation of linear forms; we introduce first the notions of null space (or kernel) and range (or column space) for matrices. The result linear f = card Basis = dim {x. f x = (0 :: b)} + dim (range f ) is more general than its corresponding version for matrices representing linear forms, since in the first one the destination vector space could be finite or infinite dimensional, whereas in its version for matrices, both the source and destination vector spaces have to be finite-dimensional. The null space correponds to the kernel of the linear form, and is a subset of the row space : definition null-space :: realˆ nˆ m => (realˆ n) set where null-space A = {x. A v x = 0 } The column space is a subset of the destination vector space of the linear form: definition col-space :: realˆ nˆ m=>(realˆ m) set where col-space A = range (λx. A v x) lemma col-space-eq: col-space A = {y. x. A v x = y} lemma null-space-eq-ker: shows null-space (matrix f ) = {x. f x = 0 } lemma col-space-eq-range: shows col-space (matrix f ) = range f After the previous equivalences between the null space and the column space and the range, the proof of the theorem for matrices is direct, as a consequence of the rank nullity theorem. lemma rank-nullity-theorem-matrices: fixes A::realˆ aˆ b shows DIM (realˆ a) = dim (null-space A) + dim (col-space A) end 4

5 References [1] S. Axler. Linear Algebra Done Right. Springer, 2nd edition, [2] M. S. Gockenbach. Finite Dimensional Linear Algebra. CRC Press,