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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.
If V and W are vector spaces, then the kernel of a linear transformation T: V → W is the set of vectors v ∈ V for which T(v) = 0. The kernel of a linear transformation is analogous to the null space of a matrix. If V is an inner product space, then the orthogonal complement to the kernel can be thought of as a generalization of the row space.
Note: This page uses common physics notation for spherical coordinates, in which is the angle between the z axis and the radius vector connecting the origin to the point in question, while is the angle between the projection of the radius vector onto the x-y plane and the x axis. Several other definitions are in use, and so care must be taken ...
They try to avoid matrix-matrix operations, but rather multiply vectors by the matrix and work with the resulting vectors. Starting with a vector , one computes , then one multiplies that vector by to find and so on. All algorithms that work this way are referred to as Krylov subspace methods; they are among the most successful methods ...
for (t 1, t 2, ... , t k) ≠ (u 1, u 2, ... , u k). [note 3] If v 1, ..., v k are linearly independent, then the coordinates t 1, ..., t k for a vector in the span are uniquely determined. A basis for a subspace S is a set of linearly independent vectors whose span is S. The number of elements in a basis is always equal to the geometric ...
In mathematics, the affine hull or affine span of a set S in Euclidean space R n is the smallest affine set containing S, [1] or equivalently, the intersection of all affine sets containing S. Here, an affine set may be defined as the translation of a vector subspace. The affine hull aff(S) of S is the set of all affine combinations of elements ...
The following are important identities in vector algebra.Identities that only involve the magnitude of a vector ‖ ‖ and the dot product (scalar product) of two vectors A·B, apply to vectors in any dimension, while identities that use the cross product (vector product) A×B only apply in three dimensions, since the cross product is only defined there.
ii) The sequence { Y n = Ker(λ i − A) n} is an increasing sequence of closed subspaces. The theorem claims it stops. Suppose it does not stop, i.e. the inclusion Ker(λ i − A) n ⊂ Ker(λ i − A) n+1 is proper for all n. By lemma 1, there exists a sequence {y n} n ≥ 2 of unit vectors such that y n ∈ Y n and d(y n, Y n − 1) > ½.