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Properties Of Vectors Addition

Check us out at Incorporating these two and many more examples in one notion of vector space is achieved via an abstract algebraic definition that disregards the concrete nature of the particular type of vectors. However, essential properties of vector addition and scalar multiplication present in the examples above are required to hold in any vector space. For example, in the algebraic example of vectors as pairs above, the result of addition does not depend on the order of the summands: (xv, yv) + (xw, yw) = (xw, yw) + (xv, yv), Likewise, in the geometric example of vectors using arrows, v + w = w + v, since the parallelogram defining the sum of the vectors is independent of the order of the vectors. To reach utmost generality, the definition of a vector space relies on the notion of a field F. A field is, essentially, a set of numbers possessing addition, subtraction, multiplication and division operations.[nb 1] Many vector spaces encountered in mathematics and sciences use the field of real numbers, but rational or complex numbers and other fields are also important. The underlying field F is fixed throughout and is specified by speaking of F-vector spaces or vector spaces over F. If F is R or C, the field of real and complex numbers, respectively, the denominations real and complex vector spaces are also common. The elements of F are called scalars. A vector space is a set V together with two operations, called vector addition and scalar multiplication. The elements of V are called vectors and are denoted in boldface.[nb 2] The sum of two vectors is denoted v + w, the product of a scalar a and a vector v is denoted a · v or av. To qualify as a vector space, addition and multiplication have to adhere to a number of requirements called axioms. They generalize properties of the vectors introduced above.[1] In the list below, let u, v, w be arbitrary vectors in V, and a, b be scalars in F. Axiom Signification Associativity of addition u + (v + w) = (u + v) + w. Commutativity of addition v + w = w + v. Identity element of addition There exists an element 0 ? V, called the zero vector, such that v + 0 = v for all v ? V. Inverse elements of addition For all v ? V, there exists an element w ? V, called the additive inverse of v, such that v + w = 0. The additive inverse is denoted ?v. Distributivity of scalar multiplication with respect to vector addition?? a(v + w) = av + aw. Distributivity of scalar multiplication with respect to field addition (a + b)v = av + bv. Compatibility of scalar multiplication with field multiplication a(bv) = (ab)v [nb 3] Identity element of scalar multiplication 1v = v, where 1 denotes the multiplicative identity in F. These axioms entail that subtraction of two vectors and division by a (non-zero) scalar can be performed via v ? w = v + (?w), v / a = (1 / a) · v. In contrast to the intuition stemming from vectors in the plane and higher-dimensional cases, there is, in general vector spaces, no notion of nearness, angles or distances. To deal with such matters, particular types of vector spaces are introduced; see below.
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