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For example, in geometry, two linearly independent vectors span a plane. To express that a vector space V is a linear span of a subset S, one commonly uses one of the following phrases: S spans V; S is a spanning set of V; V is spanned or generated by S; S is a generator set or a generating set of V.
An infinite set of vectors is linearly independent if every nonempty finite subset is linearly independent. Conversely, an infinite set of vectors is linearly dependent if it contains a finite subset that is linearly dependent, or equivalently, if some vector in the set is a linear combination of other vectors in the set.
These two vectors are said to span the resulting subspace. In general, a linear combination of vectors v 1, ... b k, v} are linearly dependent, and therefore v ∈ S.
If that is possible, then v 1,...,v n are called linearly dependent; otherwise, they are linearly independent. Similarly, we can speak of linear dependence or independence of an arbitrary set S of vectors. If S is linearly independent and the span of S equals V, then S is a basis for V.
The span of G is also the set of all linear combinations of elements of G. If W is the span of G, one says that G spans or generates W, and that G is a spanning set or a generating set of W. [12] Basis and dimension A subset of a vector space is a basis if its elements are linearly independent and span the vector space. [13]
The linear span of the -blades is called the -th exterior power of . The exterior algebra is the direct ... to be a linearly dependent set of vectors is that ...
Because the vectors usually soon become almost linearly dependent due to the properties of power iteration, methods relying on Krylov subspace frequently involve some orthogonalization scheme, such as Lanczos iteration for Hermitian matrices or Arnoldi iteration for more general matrices.
Any other pair of linearly independent vectors of R 2, such as (1, 1) and (−1, 2), forms also a basis of R 2. More generally, if F is a field , the set F n {\displaystyle F^{n}} of n -tuples of elements of F is a vector space for similarly defined addition and scalar multiplication.