What was the motivation to define what a vector space is?

In summary, the book explains that linear algebra is used to solve linear equations and that a vector space is a structure in which this is possible. Gauss' elimination works in any vector space and Fourier series decomposition for periodic functions is a good demonstration of this.
  • #1
Tosh5457
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The book I use for linear algebra explains that the motivation for defining a vector space has to do with the Gauss' reduction method taking linear combinations of the rows, but I don't understand the explanation very well. Can somebody explain?
 
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  • #2
Hi Tosh5457! :smile:

The idea behind linear algebra is to provide a theory to solve (or at least to handle) linear equations. Solving linear equation in the real numbers is quite easy using Gauss' method.

Now, a vector space is the most general possible space in which linear equations still make sense and in which they are solvable.

For example, if V is an abstract vector space, then it makes sense to ask which vectors v and w satisfy

[tex]\left\{\begin{array}{c} 2v + 3w=0\\ 3v+4w=0\\ \end{array}\right.[/tex]

This makes sense. Indeed, 2v and 3w makes sense because it is scalar multiplication (and we always have scalar multiplication on a vector space). Also 2v+3w makes sense because it's an addition (and we always have an addition on a vector space). And 0 makes sense since the vector space always has a zero. So this system of equations makes sense.

Furthermore, we can solve this system of equations by exactly the same methods as we would solve a system over [itex]\mathbb{R}[/itex]. Try to solve this system over this general vector space!

So a vector space is a structure in which it is possible to describe linear equations and in which it is possible to solve them. Also, Gauss elimination works in any vector space.

So if we want to study how to solve linear equations, we might as well study them over arbitrary vector spaces. And this is what we do.
 
  • #3
Adding to what micromass has said, a lot of phenomena act like vectors and when you have the ability to define things in term of an "inner product space", you end up getting a "geometric picture" of how particular objects behave with respect to particular basis.

Fourier series decomposition for periodic functions is a good demonstration of this. We can treat different frequency components as being independent contributions to the total signal so that no other frequency would affect a particular signal in the same way that changing the x co-ordinate of a point doesn't change the y or z co-ordinate of a point.

So with the inner product spaces (which are vector spaces with an inner product) we can actually find out orthonormal basis, and if the inner product is valid, we can treat these objects like arrows and for orthogonal basis, each contribution looks like it is at "right angles" to each other and can pictured like adding up "perpendicular lines" like a right angle triangle.

As a result, this provides a standard way of building decompositions and checking if decompositions are valid by showing orthonormality. Mathematically this is great because it is a way of breaking things down into atoms that are independent, and using this you can do all kinds of things like compressing signals, to doing fast classifications and many other applications.
 
  • #4
Thanks for the explanations :smile:
 
  • #5


The motivation for defining a vector space lies in the fundamental concept of linear combinations. In linear algebra, a linear combination is a mathematical operation that involves multiplying each element in a vector by a scalar, and then adding the resulting vectors together. This operation is essential in many areas of mathematics and science, including physics, engineering, and economics.

The Gauss' reduction method, also known as Gaussian elimination, is a powerful tool used to solve systems of linear equations. This method involves taking linear combinations of the rows of a matrix to simplify and solve the equations. In order to use this method effectively, it is crucial to have a clear understanding of what constitutes a valid linear combination and how it can be applied to different sets of vectors.

This is where the concept of a vector space comes into play. A vector space is a set of vectors that satisfy certain properties, such as closure under addition and scalar multiplication. By defining these properties, we are able to identify which sets of vectors can be used in linear combinations, and therefore, in methods like Gaussian elimination. In other words, the motivation for defining a vector space is to provide a framework for understanding and manipulating vectors in a consistent and meaningful way.

In summary, the motivation for defining a vector space is rooted in the importance of linear combinations and their applications in solving systems of linear equations. By establishing a clear and precise definition of a vector space, we are able to effectively use tools like Gaussian elimination to solve complex problems in various fields of study.
 

1. What is a vector space?

A vector space is a mathematical concept that defines a set of elements, called vectors, and a set of operations, such as addition and scalar multiplication, that can be performed on those vectors. This allows for the manipulation and analysis of quantities that have both magnitude and direction.

2. Why was the concept of a vector space defined?

The concept of a vector space was defined to provide a framework for understanding and solving problems in various fields, such as physics, engineering, and economics. By defining the properties and operations of a vector space, scientists and mathematicians can apply these concepts to real-world situations and make accurate predictions and calculations.

3. How is a vector space different from other mathematical concepts?

A vector space is different from other mathematical concepts, such as a group or a ring, because it is not limited to a specific set of numbers or operations. A vector space can be defined over any field, such as real numbers, complex numbers, or even functions. Additionally, the operations defined in a vector space are not limited to just addition and multiplication, but can also include other operations such as differentiation or integration.

4. What are some real-world applications of vector spaces?

Vector spaces have many real-world applications, such as in physics for describing the motion of objects, in computer graphics for creating and manipulating images, and in economics for modeling supply and demand. They are also used in machine learning and data analysis to represent and analyze large datasets.

5. What are the key properties of a vector space?

The key properties of a vector space include the closure under addition and scalar multiplication, the existence of a zero vector and additive inverse, and the associative and distributive properties. Other important properties include the existence of a multiplicative identity, commutativity of addition, and the ability to combine vectors to form linear combinations.

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