Table of Contents
Vector Space is a Mathematical Structure that allows us to Represent Objects as Vectors. In other words, Vector Space is a Set of Objects that can be Represented as Vectors.
Vector Space has a few Properties that allow us to Represent Objects as Vectors. These Properties are:
1. Additivity: This Property states that if we have two Vectors, and we add them together, we get a new Vector that is the Sum of the two Vectors.
2. Scalar Multiplication: This Property states that if we have a Vector and we Multiply it by a Scalar (a Number), we get a new Vector that is the Product of the Vector and the Scalar.
3. Multiplication: This Property states that if we have two Vectors, and we Multiply them together, we get a new Vector that is the Product of the two Vectors.
Vector Space Definition
In mathematics, a vector space is a set of objects called vectors, which are elements of a given vector space, together with two operations called addition and scalar multiplication. These operations allow vectors to added together and multiplied by scalars, which are real numbers.
A vector space is usually denoted by a capital letter, such as V, and the vectors in it usually denoted by lowercase letters, such as v. The two operations usually written as + and ·, so that v + w means the sum of the vectors v and w and v · w means the product of the vectors v and w.
A vector space is an important structure in mathematics, because many important mathematical objects can be described in terms of vector spaces. These include linear equations, which are equations that involve vectors, and matrices, which are rectangular arrays of numbers that can be thought of as vectors in a two-dimensional vector space.
What is Vector and Vector Space?
A vector is a mathematical object that has both magnitude and direction. Vectors often represented using arrows in two or three dimensional space. A vector space is a set of all vectors in a given dimensional space.
Vector Addition
Vector addition is the process of adding two or more vectors together to produce a new vector. The resultant vector is the sum of the individual vectors, with the direction and magnitude of the resultant vector determined by the individual vectors’ directions and magnitudes.
To calculate the resultant vector, we use the following formula:
\vec{R} = \vec{A} + \vec{B}
Where \(\vec{A}\) and \(\vec{B}\) are the individual vectors and \(\vec{R}\) is the resultant vector.
Example
Calculate the resultant vector for the following two vectors:
\vec{A} = (3, 2)
\vec{B} = (4, 1)
We first calculate the magnitude of each vector:
\vec{A} = (3, 2)
\vec{A} \times \vec{A} = 9
\vec{B} = (4, 1)
\vec{B} \times \vec{B} = 16
Next, we calculate the direction of each vector:
\vec{A} = (3, 2)
vec{A} \times \vec{A} = 9
\vec{A} \times \vec{B} = -2
\vec{B} = (4, 1)
Scalar Multiplication
A scalar is a number, and multiplication is just multiplying two numbers together. So scalar multiplication is just multiplying two numbers together.
Axioms for Vector Spaces
A vector space is a set of objects, called vectors, that have a common mathematical structure. The vectors in a vector space usually thought of as points in space, but they can also thought of as objects with other mathematical properties.
There are a few basic properties that all vectors in a vector space have in common. These properties called axioms.
The axioms for a vector space are:
1. Addition: Vectors can added together to create a new vector. The result is a vector that is the sum of the original vectors.
2. Scalar Multiplication: A vector can multiplied by a number, called a scalar. The result is a vector that is the product of the original vector and the scalar.
3. Multiplication: Vectors can multiplied together to create a new vector. The result is a vector that is the product of the original vectors.
4. Associative Property: When vectors multiplied together, the order of multiplication does not affect the result.
5. Distributive Property: When vectors multiplied by scalars, the scalars distributed over the vectors.
Vector Space Examples
Let’s take a look at some examples of vector spaces.
The following are all vector spaces:
- set of all real numbers
- The set of all complex numbers
- set of all polynomials of degree n
- The set of all ordered pairs of real numbers
- set of all vectors in a given dimension
- The set of all functions from a given set to another set
