Sage-Tips

What is the difference between a group and a vector space?

This means that if you take V and remove the scalar multiplication operator, the elements of V forms a group and commute with each others. Conversely, you can take an abelian group and try to turn it into a vector space by adding scalar multiplication on it.

What is the mathematical basis of vector?

In mathematics, a set B of vectors in a vector space V is called a basis if every element of V may be written in a unique way as a finite linear combination of elements of B. The coefficients of this linear combination are referred to as components or coordinates of the vector with respect to B.

What is general vector space?

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.

Is every group a vector space?

First of all, a vector space is an abelian group, with the operation being addition. Second of all, you have another operation called scalar multiplication, whereby you can take an element of your vector space and multiply it by an element of your field of scalars (normally either the real or complex numbers).

Is the general linear group a vector space?

Real case. The general linear group GL(n, R) over the field of real numbers is a real Lie group of dimension n2. To see this, note that the set of all n×n real matrices, Mn(R), forms a real vector space of dimension n2.

What is the difference between A and a vector?

Explanation: A scalar quantity has magnitude but no direction. An example of a scalar quantity is density. A vector quantity has both magnitude and direction.

Does a vector space always have a basis?

Summary: Every vector space has a basis, that is, a maximal linearly inde- pendent subset. Every vector in a vector space can be written in a unique way as a finite linear combination of the elements in this basis.

What is the most general definition of a vector?

Prelude: A vector, as defined below, is a specific mathematical structure. It has numerous physical and geometric applications, which result mainly from its ability to represent magnitude and direction simultaneously. Vectors are quite useful in simplifying problems from three-dimensional geometry.

Is an algebra a vector space?

In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. The ring of real square matrices of order n forms a unital algebra since the identity matrix of order n is the identity element with respect to matrix multiplication.

Can vector be compared?

A vector quantity has two characteristics, a magnitude and a direction. When comparing two vector quantities of the same type, you have to compare both the magnitude and the direction. On this slide we show three examples in which two vectors are being compared. Vectors are usually denoted on figures by an arrow.

What is a vector space in math?

Vector Space A vector space or a linear space is a group of objects called vectors, added collectively and multiplied (“scaled”) by numbers, called scalars. Scalars are usually considered to be real numbers. But there are few cases of scalar multiplication by rational numbers, complex numbers, etc. with vector spaces.

What is the difference between the definition of vector space and group?

When comparing the difference between the definition of vector space, I see that the main job is that vector space defines a scalar product while the group not, so here list two of my questions? 1… Stack Exchange Network

What is the difference between real and complex 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.

What is the difference between linear subspace and vector space?

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.