Linear Algebra Vector Spaces

A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars. Scalars are mostly called true numbers, but vector spaces with scalar multiplication by complex numbers, rational numbers, or typically a field are also present. The operations involving vector addition and scalar multiplication must follow certain criteria known as vector axioms. The terms real vector space and complex vector space are also used to determine that the scalars are real or difficult numbers. Some sets of Euclidean vectors are common vector space examples. They are physical numbers, such as forces, in which a third may be added to generate any two forces (of the same type), and the multiplication of an actual multiplier by a force vector is a second force vector. Vectors representing flat or tridimensional space shifts often form vector spaces in the same vein (but more geometrically). In vector spatial space vectors need not necessarily be arrow-like objects, as they appear in the above examples: vectors are known as abstract mathematical objects with unique properties, which can be interpreted as arrows in certain cases. One of the fundamental branches of mathematics is linear algebra. In many practical problems, quantities to be measured are connected to measurable quantities by linear equations and can thus be evaluated using the traditional row operating methods. In many cases the methods to be used are the most obvious (and most important). Further refinement of the theory's vector space leads to methods of solving linear differential equations, and even necessarily nonlinear equations are typically approached by a repeated solution of the required linearly equation in order to find a convergent sequence of approximate solutions. Vector space is a vector set V which has two operations and is called addition and scalar propagation which is called vector objects. In addition, in many other branches of mathematics the theory of the vector spaces originally generated to solve linear equations is widespread and built on. Historically, the first ideas which have led to vectors can be traced to the analytical geometry, matrices, systems of linear equations and Euclidean vectors of the 17th century. In 1888, Giuseppe Peano proposed his new, more abstract study, involving more general objects than the Euclidean space; however, most of the theories can be considered an extension of classical geometric ideas such as lines, planes and higher-dimensional analogues. The concept of vector space will first be explained by describing two particular examples:

The first definition of a vector space is a fixed plane with arrows beginning at a specified point. This is the explanation of forces or speeds in physics. Given the two flips, v and w, there is one diagonal arrow in the parallelogram of the two arrows, which also begins at the origin. In the case of two arrows on the same side, the sum of this new flat is the arrow with a length that corresponds to the total or difference in length, depending on whether or not the arrows have the same direction. This flat is the total of both arrows and the length of the arrow. [1]

as v, is called the multiplication of v by a, because of its length. It's referred to as av. Av is defined, if an is negative, as the flew pointing the other way. The following shows a few examples: if a = 2, the resulting vector aw has the same direction as w, but is stretched to the double length of w (right image below). Equivalently, 2w is the sum w + w. Moreover, (−1)v = −v has the opposite direction and the same length as v (blue vector pointing down in the right image).

The pairs of real numbers x and y are a second key example of a vector space. (The sequence is important for the components x and y, so that such a pair has been often referred to as the ordered pair.) The sum of two such pairs and multiplication of a pair with a number is defined as follows: A vector space over a field F is a set V together with two operations that satisfy the eight axioms listed below. In the following, V × V denotes the Cartesian product of V with itself, and → denotes a mapping from one set to another. ► The first operation, called vector addition or simply addition + : V × V → V, takes any two vectors v and w and assigns to them a third vector which is commonly written as v + w, and called the sum of these two vectors. (The resultant vector is also an element of the set V.) ► The second operation, called scalar multiplication · : F × V → V， takes any scalar a and any vector v and gives another vector av. (Similarly, the vector AV is a V set unit. It should not be confused with the scalar product, also known as the internal product or point product, which has an additional structure in some unique vector spaces but does not exist in all of them. Scalar multiplication is a scalar multiplication of a vector; the other means two vectors which generate a scalar.) The field is the field of actual numbers in the two examples above, and the set of vectors is composed of planar arrows with fixed starting point and pairs of real numbers. The set V and the add-in / multiplication operations must comply with several criteria called axioms to be considered a vector space[2]. These are shown in the following table, with u, v and w indicating arbitrary V vectors and b and a denoting scalar of F. Indeed, the result of addition of two ordered pairs (as in the second example above) does not depend on the order of the summands: In the geometric example, v + w = w + v as the vector‟s space total parallelogram is not in the order of the vectores. v + w = w + v. In both cases, all other axioms can be tested similarly. Thus, the description combines these two and many more examples in one notion of vector space by disregarding the unique existence of the particular form of vector space. Subtraction of two vectors and division by a (non-zero) scalar can be defined as The vector space is called a true vector space if the scalar F is the real R number. The vector space is called complex vector space if the scalar field is the complex C number. The two cases in the engineering industry are most common. The overall vector space concept requires the scalar elements to be part of a fixed F field. The word F-vector space or vector space over F is then defined. A field is basically a set of numbers with operations to add, deduct, multiply and divide. Rational numbers, for instance, form a field. There is no concept of proximity, angles or lengths, as opposed to the vector intuition in the plane and higher dimensional situations. Special forms of vector spaces are implemented for such matters; see § Vector spaces for more information below with additional structures.

A vector space is a non-empty set V, whose objects are referred to as vectors, fitted with two operations called addition and scalar multiplication: there are uniform vectors u + v and cu in V of each of the two properties. By definition of vector space it is easy to see that for any vector u and scalar c, 0u = 0, c0 = 0, −u = (−1)u. For instance, Example 1. (a) The Euclidean space R n is a vector space under the ordinary addition and scalar multiplication. (b) The set Pn of all polynomials of degree less than or equal to n is a vector space under the ordinary addition and scalar multiplication of polynomials. (c) The set M(m, n) of all m × n matrices is a vector space under the ordinary addition and scalar multiplication of matrices. (d) The set C[a, b] of all continuous functions on the closed interval [a, b] is a vector space under the ordinary addition and scalar multiplication of functions.

(a) For a vector space V, the set {0} of the zero vector and the whole space V are subspaces of V; they are called the trivial subspaces of V . (b) For an m × n matrix A, the set of solutions of the linear system Ax = 0 is a subspace of Rn. However, if b 6= 0, the set of solutions of the system Ax = b is not a subspace of Rn. (c) For any vectors v1, v2, . . . , vk in Rn, the span Span {v1, v2, . . . , vk} is a subspace of R n. image of T is a subspace of Rm, and the inverse image is a subspace of Rn.

Scalar multiplication are vector additions and scalar multiplication, which satisfy the closure properties: u + v and av are in V for all an in F, and u, v in V. In the abstract algebra language, the first four axioms are analogous to the fact that the vector set should be an abelian group. The other axioms include a structure of this category with an F-module. In other words, the ring homomorphism f from field F into the vector group endomorphism ring. The scalar multiplication av, then (f(a))(v) is defined[6]. The vector space axioms have some direct implications. Some come from the basic group principle, applied to the vector group additive. For example, the 0 vector of 0 is unique and the reverse additive v of any vector v is unique. Additional properties follow those for scalar multiplication, for example av is equal to 0 if and only if the distributive law is equal to 0 or v.

The set of complex numbers C, that is, numbers that can be written in the form x + iy for real numbers x and y where i is the imaginary unit, form a vector space over the reals with the usual addition and multiplication: for real numbers x, y, a, b and c. The different axioms of a vector space are based on the assumption that the same laws apply to complex arithmetic numbers. Indeed, the definition of a complex number (i.e., isomorphically) is basically the same for the vector space of the ordained actual number pairs described previously: when we consider the complex number x + iy to be the ordered pair (x , y) in the complex plane, we can see that there is a direct connexion between the additions and the scale-producing rules of the above case. Field extensions typically provide another class of vector space examples, in algebra and algebraic

example, the complex numbers are a vector space over , and the field extension is a vector space over Q.

Functions from any fixed set Ω to a field F also form vector spaces, by performing addition and scalar multiplication point wise. That is, the sum of two functions f and g is the function (f + g) given by and similarly for multiplication. In certain geometric situations such functional spaces occur when a ren is the actual line, an interval, or other sub-sets of R. Various principles in topology and analysis, such as continuity, integrability or differentiability, are well preserved as regards linearity: the sums and scalar multiples of functions with this sort of property still have the property [17]. The methods of functional analysis, see below, are discussed in more detail. Algebraic constraints also yield vector spaces: the vector space F[x] is given by polynomial functions: where the coefficients

Main articles: Linear equation, linear differential equation, and Systems of linear equations Systems of homogeneous linear equations are closely tied to vector spaces. For example, the solutions of solutions, and yet satisfy the same proportion of the three variables. Matrices can be used for condensing several linear equations into a single vector equation as above, namely Ax = 0, Where x is a vectour (a, b , c), Ax denotes the sum of the matrix and 0 = (0, 0) is the zero vector. Ax is the matrix of the equations. The solutions of uniform linear differential equations, in a similar way, form vector spaces. For example, yields where a and b are arbitrary constants, and ex is the natural exponential function.

In linear map or linear transformation can express the relationship of two vector spaces. They represent the space structure vector, that is to say, maintain amounts and scalar multiplication: for all v and w in V, all a in F. An isomorphism is a linear map f : V → W such that there exists an inverse map g : W → V, which is a map such that the two possible compositions f ∘ g : W → W and g ∘ f : V → V are identity maps. Equivalently, f is both one-to-one (injective) and onto (surjective). If an isomorphism occurs between V and W, both spaces would be isomorphic and are basically identical as vector spaces since all identities containing V are transported via V to similar spaces in W and vice versa via g. For instance, the input "plane arrows" and "ordered numbers pairs" are isomorphic: the x- and y-component of the arrow can be taken into consideration by the planned arrow v which starts at the beginning of the (fixed-) co-ordinate system in an ordered pair. In comparison, if a pair (x , y) is seen, the arrown is x to the right and y up (down, where y is negative) to x to the left. Linear maps V → W between two vector spaces form a vector space HomF(V, W), also denoted L(V, W). The space of linear maps from V to F is called the dual vector V∗∗, any vector space can be embedded into its bidual; the map is an isomorphism if and only if the space is finite-dimensional.

Once a basis of V is chosen, linear maps f : V → W are completely determined by specifying the images of the basis vectors, because any element of V is expressed uniquely as a linear combination of them. If dim V = dim W, a 1-to-1 correspondence between fixed bases of V and W gives rise to a linear map that maps any basis element of V to the corresponding basis element of W. It is an isomorphism, by its very definition. Therefore, two vector spaces are isomorphic if their dimensions agree and vice versa. Another way to express this is that any vector space is completely classified (up to isomorphism) by its dimension, a single number. In particular, any n-dimensional F-vector space V is isomorphic to Fn. There is, however, no "canonical" or preferred isomorphism; actually an isomorphism φ : Fn → V is equivalent to the choice of a basis of V, by mapping the standard basis of Fn to V, via φ.

Vector spaces stem from affine geometry, via the introduction of coordinates in the plane or three-dimensional space. By defining solutions to an equation of two variables with points on a plane curve, mathematicians René Descartes and Pierre de Fermat created analytical geometry. Vectors were checked when Argand and Hamilton introduced complex numbers and the latter started quaternions. These are R2 and R4 elements; they are handled using linear combinations and specified linear equation systems. There are definitions of linear autonomy and dimension as well as scalar products. The 1844 work of Grassmann actually goes beyond the vectors, because his careful reproduction often leads him to what today are referred to as algebras. The first to give a modern description of vector spaces and linear maps was the Italian mathematician Peano. The construction of functional spaces by Henri Lebesgue is a significant advancement of vector spaces. Algebra and the new area of functional analysis began interacting at the integrable spaces and Hilbert spaces. The first experiments on infinite dimensional vector spaces were also carried out at this time.

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Assistant Professor, Mathematics Dayanand Science College, Latur