Search results
Results from the WOW.Com Content Network
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.
This article uses the convention that vectors are denoted in a bold font (e.g. a 1), and scalars are written in normal font (e.g. a 1). The dot product of vectors a and b is written as a ⋅ b {\displaystyle \mathbf {a} \cdot \mathbf {b} } , the norm of a is written ‖ a ‖, the angle between a and b is denoted θ .
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.
It is positive definite: for all vectors x, x ⋅ x ≥ 0 , with equality if and only if x = 0. An operation on pairs of vectors that, like the dot product, satisfies these three properties is known as a (real) inner product. A vector space equipped with such an inner product is known as a (real) inner product space.
The concept of orthogonality may be extended to a vector space over any field of characteristic not 2 equipped with a quadratic form .Starting from the observation that, when the characteristic of the underlying field is not 2, the associated symmetric bilinear form , = ((+) ()) allows vectors and to be defined as being orthogonal with respect to when (+) () = .
The row vectors of a matrix.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.
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]
But often, it is easier to deal with vectors of unit length. That is, it often simplifies things to only consider vectors whose norm equals 1. The notion of restricting orthogonal pairs of vectors to only those of unit length is important enough to be given a special name. Two vectors which are orthogonal and of length 1 are said to be orthonormal.