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The dimension of the row space is called the rank of the matrix. This is the same as the maximum number of linearly independent rows that can be chosen from the matrix, or equivalently the number of pivots. For example, the 3 × 3 matrix in the example above has rank two. [9] The rank of a matrix is also equal to the dimension of the column space.
Rank–nullity theorem. The rank–nullity theorem is a theorem in linear algebra, which asserts: the number of columns of a matrix M is the sum of the rank of M and the nullity of M; and; the dimension of the domain of a linear transformation f is the sum of the rank of f (the dimension of the image of f) and the nullity of f (the dimension of ...
The nullity of M is given by m − n + c, where, c is the number of components of the graph and n − c is the rank of the oriented incidence matrix. This name is rarely used; the number is more commonly known as the cycle rank, cyclomatic number, or circuit rank of the graph. It is equal to the rank of the cographic matroid of the graph.
A -finite measure on a measurable space (,) is called atomic or purely atomic if every measurable set of positive measure contains an atom. This is equivalent to say that there is a countable partition of formed by atoms up to a null set. [3]
A partially ordered set with least element 0 is called atomistic (not to be confused with atomic) if every element is the least upper bound of a set of atoms. The linear order with three elements is not atomistic (see Fig. 2). Atoms in partially ordered sets are abstract generalizations of singletons in set theory (see Fig. 1). Atomicity (the ...
As a result, there can be only one set with no elements, hence the usage of "the empty set" rather than "an empty set". The only subset of the empty set is the empty set itself; equivalently, the power set of the empty set is the set containing only the empty set. The number of elements of the empty set (i.e., its cardinality) is zero. The ...
The Oddo–Harkins rule holds that an element with an even atomic number is more abundant than the elements with immediately adjacent atomic numbers. For example, carbon, with atomic number 6, is more abundant than boron (5) and nitrogen (7). Generally, the relative abundance of an even atomic numbered element is roughly two orders of magnitude ...
E may be called the ground set of M. Its elements may be called the points of M. A subset of E spans M if its closure is E. A set is said to span a closed set K if its closure is K. The girth of a matroid is the size of its smallest circuit or dependent set. An element that forms a single-element circuit of M is called a loop. Equivalently, an ...