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Class 10 Maths Notes for FBISE by ClassNotes - All Chapters A vector space (also called a linear space) is a set of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called myboat279 boatplanss are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any myboat279 boatplans operations of vector addition and scalar multiplication must. Develop apps that help serve the CH community Paid to refer users to affiliate programs and services Artists, Activists, Authors, Actors, Billionaires, and Health leaders in the world. Million YouTube (+ million views) Million Instagram Million Facebook k TikTok Subscribe to The School of Greatness at lewishowes. Objective Proficiency. Student's Book 2ed, p. Pages. Objective Proficiency. Student's Book 2ed, p.
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This article deals mainly with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces. The first example of a vector space consists of arrows in a fixed plane , starting at one fixed point. This is used in physics to describe forces or velocities. Given any two such arrows, v and w , the parallelogram spanned by these two arrows contains one diagonal arrow that starts at the origin, too.

Another operation that can be done with arrows is scaling: given any positive real number a , the arrow that has the same direction as v , but is dilated or shrunk by multiplying its length by a , is called multiplication of v by a. It is denoted a v. When a is negative, a v is defined as the arrow pointing in the opposite direction instead.

A second key example of a vector space is provided by pairs of real numbers x and y. The Class 10 Maths Ch 1 Ex 1.1 Paper order of the components x and y is significant, so such a pair is also called an ordered pair. Such a pair is written as x , y. The sum of two such pairs and multiplication of a pair with a number is defined as follows:.

The first example above reduces to this one, if the arrows are represented by the pair of Cartesian coordinates of their endpoints. In this article, vectors are represented in boldface to distinguish them from scalars. A vector space over a field F is a set V together with two operations that satisfy the eight axioms listed below. Elements of V are commonly called vectors. Elements of F are commonly called scalars. Common symbols for denoting vector spaces include U , Class 10 Maths Ch 1 Ex 1.1 Library V , and W.

In the two examples above, the field is the field of the real numbers, and the set of the vectors consists of the planar arrows with fixed starting point and pairs of real numbers, respectively.

To qualify as a vector space, the set V and the operations of addition and multiplication must adhere to a number of requirements called axioms. These axioms generalize properties of the vectors introduced in the above examples. Indeed, the result of addition of two ordered pairs as in the second example above does not depend on the order of the summands:. All other axioms can be verified in a similar manner in both examples.

Thus, by disregarding the concrete nature of the particular type of vectors, the definition incorporates these two and many more examples in one notion of vector space. When the scalar field F is the real numbers R , the vector space is called a real vector space.

When the scalar field is the complex numbers C , the vector space is called a complex vector space. These two cases are the ones used most often in engineering. The general definition of a vector space allows scalars to be elements of any fixed field F. The notion is then known as an F - vector space or a vector space over F. A field is, essentially, a set of numbers possessing addition , subtraction , multiplication and division operations. In contrast to the intuition stemming from vectors in the plane and higher-dimensional cases, in general vector spaces, there is no notion of nearness , angles or distances.

Some older sources mention these properties as separate axioms. In the parlance of abstract algebra , the first four axioms are equivalent to requiring the set of vectors to be an abelian group under addition. The remaining axioms give this group an F - module structure. In other words, there is a ring homomorphism f from the field F into the endomorphism ring of the group of vectors. Then scalar multiplication a v is defined as f a v.

There are a number of direct consequences of the vector space axioms. Further properties follow by employing also the distributive law for the scalar multiplication, for example a v equals 0 if and only if a equals 0 or v equals 0. Vector spaces stem from affine geometry , via the introduction of coordinates in the plane or three-dimensional space. Bellavitis introduced the notion of a bipoint, i. In , Cayley introduced the matrix notation which allows for a harmonization and simplification of linear maps.

He envisaged sets of abstract objects endowed with operations. Actually Grassmann's work exceeds the framework of vector spaces, since his considering multiplication, too, led him to what are today called algebras.

Italian mathematician Peano was the first to give the modern definition of vector spaces and linear maps in An important development of vector spaces is due to the construction of function spaces by Henri Lebesgue.

This was later formalized by Banach and Hilbert , around The simplest example of a vector space over a field F is the field itself, equipped with its standard addition and multiplication.

More generally, all n -tuples sequences of length n. The various axioms of a vector space follow from the fact that the same rules hold for complex number arithmetic.

More generally, field extensions provide another class of examples of vector spaces, particularly in algebra and algebraic number theory : a field F containing a smaller field E is an E -vector space, by the given multiplication and addition operations of F.

Many notions in topology and analysis, such as continuity , integrability or differentiability are well-behaved with respect to linearity: sums and scalar multiples of functions possessing such a property still have that property.

They are studied in greater detail using the methods of functional analysis , see below. Systems of homogeneous linear equations are closely tied to vector spaces. They form a vector space: sums and scalar multiples of such triples still satisfy the same ratios of the three variables; thus they are solutions, too.

Matrices can be used to condense multiple linear equations as above into one vector equation, namely. In a similar vein, the solutions of homogeneous linear differential equations form vector spaces.

For example,. Bases allow one to represent vectors by a sequence of scalars called coordinates or components. Linear independence means that the coordinates a k are uniquely determined for any vector in the vector space. The corresponding coordinates x 1 , x 2 , � , x n are just the Cartesian coordinates of the vector. Every vector space has a basis. This follows from Zorn's lemma , an equivalent formulation of the Axiom of Choice. Dimension theorem for vector spaces. If the space is spanned by finitely many vectors, the above statements can be proven without such fundamental input from set theory.

The dimension of the coordinate space F n is n , by the basis exhibited above. The dimension of the polynomial ring F [ x ] introduced above [ clarification needed ] is countably infinite , a basis is given by 1 , x , x 2 , � A fortiori , the dimension of more general function spaces, such as the space of functions on some bounded or unbounded interval, is infinite. These two functions are linearly independent over R , so the dimension of this space is two, as is the degree of the equation.

A field extension over the rationals Q can be thought of as a vector space over Q by defining vector addition as field addition, defining scalar multiplication as field multiplication by elements of Q , and otherwise ignoring the field multiplication. Thus, C is a two-dimensional R -vector space and, as any field, one-dimensional as a vector space over itself, C. The relation of two vector spaces can be expressed by linear map or linear transformation.

They are functions that reflect the vector space structure, that is, they preserve sums and scalar multiplication:. Equivalently, f is both one-to-one injective and onto surjective. For example, the "arrows in the plane" and "ordered pairs of numbers" vector spaces in the introduction are isomorphic: a planar arrow v departing at the origin of some fixed coordinate system can be expressed as an ordered pair by considering the x - and y -component of the arrow, as shown in the image at the right.

Conversely, given a pair x , y , the arrow going by x to the right or to the left, if x is negative , and y up down, if y is negative turns back the arrow v. It is an isomorphism, by its very definition. 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 F n. The freedom of choosing a convenient basis is particularly useful in the infinite-dimensional context; see below. Matrices are a useful notion to encode linear maps. Any m -by- n matrix A gives rise to a linear map from F n to F m , by the following. The determinant det A of a square matrix A is a scalar that tells whether the associated map is an isomorphism or not: to be so it is sufficient and necessary that the determinant is nonzero.

The vector space V may or may not possess an eigenbasis , a basis consisting of eigenvectors. This phenomenon is governed by the Jordan canonical form of the map. To achieve the spectral theorem , the corresponding statement in the infinite-dimensional case, the machinery of functional analysis is needed, see below.

In addition to the above concrete examples, there are a number of standard linear algebraic constructions that yield vector spaces related to given ones.

In addition to the definitions given below, they are also characterized by universal properties , which determine an object X by specifying the linear maps from X to any other vector space. A nonempty subset W of a vector space V that is closed under addition and scalar multiplication and therefore contains the 0 -vector of V is called a linear subspace of V , or simply a subspace of V , when the ambient space is unambiguously a vector space.

The intersection of all subspaces containing a given set S of vectors is called its span , and it is the smallest subspace of V containing the set S. Expressed in terms of elements, the span is the subspace consisting of all the linear combinations of elements of S.

A linear subspace of dimension 1 is a vector line. A linear subspace of dimension 2 is a vector plane.

A linear subspace that contains all elements but one of a basis of the ambient space is a vector hyperplane. In a vector space of finite dimension n , a vector hyperplane is thus a subspace of dimension n � 1. The counterpart to subspaces are quotient vector spaces. This concept also extends to linear differential equations. The direct product of vector spaces and the direct sum of vector spaces are two ways of combining an indexed family of vector spaces into a new vector space.

If the index set I is finite, the two constructions agree, but in general they are different. The tensor product is a particular vector space that is a universal recipient of bilinear maps g , as follows. It is defined as the vector space consisting of finite formal sums of symbols called tensors. From the point of view of linear algebra, vector spaces are completely understood insofar as any vector space is characterized, up to isomorphism, by its dimension.

However, vector spaces per se do not offer a framework to deal with the question�crucial to analysis�whether a sequence of functions converges to another function. Likewise, linear algebra is not adapted to deal with infinite series , since the addition operation allows only finitely many terms to be added.

Therefore, the needs of functional analysis require considering additional structures. Ordered vector spaces , for example Riesz spaces , are fundamental to Lebesgue integration , which relies on the ability to express a function as a difference of two positive functions. Vector spaces endowed with such data are known as normed vector spaces and inner product spaces , respectively. Coordinate space F n can be equipped with the standard dot product :.

In R 2 , this reflects the common notion of the angle between two vectors x and y , by the law of cosines :. An important variant of the standard dot product is used in Minkowski space : R 4 endowed with the Lorentz product. Singling out the fourth coordinate� corresponding to time , as opposed to three space-dimensions�makes it useful for the mathematical treatment of special relativity. Convergence questions are treated by considering vector spaces V carrying a compatible topology , a structure that allows one to talk about elements being close to each other.

In such topological vector spaces one can consider series of vectors. The infinite sum. For example, the f i could be real or complex functions belonging to some function space V , in which case the series is a function series. The mode of convergence of the series depends on the topology imposed on the function space. In such cases, pointwise convergence and uniform convergence are two prominent examples.

A way to ensure the existence of limits of certain infinite series is to restrict attention to spaces where any Cauchy sequence has a limit; such a vector space is called complete. Roughly, a vector space is complete provided that it contains all necessary limits. For example, the vector space of polynomials on the unit interval [0,1], equipped with the topology of uniform convergence is not complete because any continuous function on [0,1] can be uniformly approximated by a sequence of polynomials, by the Weierstrass approximation theorem.

Banach and Hilbert spaces are complete topological vector spaces whose topologies are given, respectively, by a norm and an inner product. Their study�a key piece of functional analysis �focuses on infinite-dimensional vector spaces, since all norms on finite-dimensional topological vector spaces give rise to the same notion of convergence. In the infinite-dimensional case, however, there will generally be inequivalent topologies, which makes the study of topological vector spaces richer than that of vector spaces without additional data.

From a conceptual point of view, all notions related to topological vector spaces should match the topology. The fundamental Hahn�Banach theorem is concerned with separating subspaces of appropriate topological vector spaces by continuous functionals. Banach spaces , introduced by Stefan Banach , are complete normed vector spaces. These spaces are complete. Imposing boundedness conditions not only on the function, but also on its derivatives leads to Sobolev spaces.

Complete inner product spaces are known as Hilbert spaces , in honor of David Hilbert. By definition, in a Hilbert space any Cauchy sequence converges to a limit.

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