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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.}
Linear span Given a subset G of a vector space V, the linear span or simply the span of G is the smallest linear subspace of V that contains G, in the sense that it is the intersection of all linear subspaces that contain G. The span of G is also the set of all linear combinations of elements of G.
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 ...
If V is a vector space over a field K, a subset W of V is a linear subspace of V if it is a vector space over K for the operations of V.Equivalently, a linear subspace of V is a nonempty subset W such that, whenever w 1, w 2 are elements of W and α, β are elements of K, it follows that αw 1 + βw 2 is in W.
Let the field K be the set R of real numbers, and let the vector space V be the Euclidean space R 3. Consider the vectors e 1 = (1,0,0), e 2 = (0,1,0) and e 3 = (0,0,1). Then any vector in R 3 is a linear combination of e 1, e 2, and e 3. To see that this is so, take an arbitrary vector (a 1,a 2,a 3) in R 3, and write:
The row space, or coimage, of a matrix A is the span of the row vectors of A. By the above reasoning, the kernel of A is the orthogonal complement to the row space. That is, a vector x lies in the kernel of A, if and only if it is perpendicular to every vector in the row space of A.
If we go on to Hilbert spaces, a non-orthonormal set of vectors having the same linear span as an orthonormal basis may not be a basis at all. For instance, any square-integrable function on the interval [ − 1 , 1 ] {\displaystyle [-1,1]} can be expressed ( almost everywhere ) as an infinite sum of Legendre polynomials (an orthonormal basis ...
The vector projection of a vector on a nonzero vector is defined as [note 1] = , , , where , denotes the inner product of ... and the span of ...