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The sum and product of finite sequences of real numbers.
 Formalized Mathematics,
, 1990
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The Euclidean Space
, 1991
"... this paper. In this paper k, n are natural numbers and r is a real number. Let us consider n. The functor R ..."
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this paper. In this paper k, n are natural numbers and r is a real number. Let us consider n. The functor R
The product and the determinant of matrices with entries in a field
, 2003
"... Concerned with a generalization of concepts introduced in [14], i.e. there are introduced the sum and the product of matrices of any dimension of elements of any field. ..."
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Concerned with a generalization of concepts introduced in [14], i.e. there are introduced the sum and the product of matrices of any dimension of elements of any field.
Semigroup operations on finite subsets
 Journal of Formalized Mathematics
, 1990
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Associated Matrix of Linear Map
, 2002
"... this paper. 1. PRELIMINARIES Let A be a set, let X be a set, let D be a non empty set of finite sequences of A, let p be a partial function from X to D, and let i be a set. Then p i is an element of D ..."
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this paper. 1. PRELIMINARIES Let A be a set, let X be a set, let D be a non empty set of finite sequences of A, let p be a partial function from X to D, and let i be a set. Then p i is an element of D
Sum and product of finite sequences of elements of a field
 Journal of Formalized Mathematics
, 1992
"... Summary. This article is concerned with a generalization of concepts introduced in [11], i.e., there are introduced the sum and the product of finite number of elements of any field. Moreover, the product of vectors which yields a vector is introduced. According to [11], some operations on ituples ..."
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Summary. This article is concerned with a generalization of concepts introduced in [11], i.e., there are introduced the sum and the product of finite number of elements of any field. Moreover, the product of vectors which yields a vector is introduced. According to [11], some operations on ituples of elements of field are introduced: addition, subtraction, and complement. Some properties of the sum and the product of finite number of elements of a field are present.
Homomorphisms of algebras. Quotient universal algebra
 Formalized Mathematics
, 1993
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The rank+nullity theorem
 Formalized Mathematics
"... Summary. The rank+nullity theorem states that, if T is a linear transformation from a finitedimensional vector space V to a finitedimensional vector space W, then dim(V) = rank(T) + nullity(T), where rank(T) = dim(im(T)) and nullity(T) = dim(ker(T)). The proof treated here is standard; see, for ..."
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Summary. The rank+nullity theorem states that, if T is a linear transformation from a finitedimensional vector space V to a finitedimensional vector space W, then dim(V) = rank(T) + nullity(T), where rank(T) = dim(im(T)) and nullity(T) = dim(ker(T)). The proof treated here is standard; see, for example, [14]: take a basis A of ker(T) and extend it to a basis B of V, and then show that dim(im(T)) is equal to B − A, and that T is onetoone on B − A.