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In mathematics, the linear span (also called the linear hull [1] or just span) of a set of elements of a vector space is the smallest linear subspace of that contains . It is the set of all finite linear combinations of the elements of S , [ 2 ] and the intersection of all linear subspaces that contain S . {\displaystyle S.}
In mathematics and physics, vector notation is a commonly used notation for representing vectors, [1] [2] which may be Euclidean vectors, or more generally, members of a vector space. For denoting a vector, the common typographic convention is lower case, upright boldface type, as in v .
In this article, vectors are represented in boldface to distinguish them from scalars. [nb 1] [1] A vector space over a field F is a non-empty set V together with a binary operation and a binary function that satisfy the eight axioms listed below. In this context, the elements of V are commonly called vectors, and the elements of F are called ...
The row space of this matrix is the vector space spanned by the row vectors. The column vectors of a matrix. The column space of this matrix is the vector space spanned by the column vectors. In linear algebra, the column space (also called the range or image) of a matrix A is the span (set of all possible linear combinations) of its column ...
A set of vectors is linearly independent if none is in the span of the others. Equivalently, a set S of vectors is linearly independent if the only way to express the zero vector as a linear combination of elements of S is to take zero for every coefficient a i. A set of vectors that spans a vector space is called a spanning set or generating set.
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the linear span in a vector space (also often denoted Span(S)), the generated subgroup in a group, the generated ideal in a ring, the generated submodule in a module. 2. Often used, mainly in physics, for denoting an expected value. In probability theory, () is generally used instead of .