Difference between an Algebra and a Vector space

In summary, the conversation discusses the difference between an algebra and a vector space. One distinction is that an algebra has a second binary operation, namely multiplication, in addition to the vector space structure. Any further guidance on this topic is appreciated.
  • #1
Milsomonk
96
17
Hi guys,
I am a little confused on the difference between an Algebra and a vector space. I´m guessing there´s a fairly simple distinction. Any guidance would be much appreciated, I accept both hand wavy and hardcore axiom based responses :)
Cheers
 
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  • #2
Milsomonk said:
Hi guys,
I am a little confused on the difference between an Algebra and a vector space. I´m guessing there´s a fairly simple distinction. Any guidance would be much appreciated, I accept both hand wavy and hardcore axiom based responses :)
Cheers
An algebra is a vector space, on which additionally a second binary operation, usually referred to as multiplication is defined.
 
  • #3
Amazing, thanks :)
 

Related to Difference between an Algebra and a Vector space

What is the definition of an algebra?

An algebra is a mathematical structure that consists of a set of elements, operations, and rules for combining these elements through the operations. It is used to study and solve equations, manipulate symbols, and understand the properties of numbers and other mathematical objects.

What is the definition of a vector space?

A vector space is a mathematical structure that consists of a set of vectors, operations, and rules for combining these vectors through the operations. It is used to represent and manipulate physical quantities, such as velocity and force, and to study the properties of geometric objects.

What is the main difference between an algebra and a vector space?

The main difference between an algebra and a vector space is the type of elements they contain. Algebras contain elements that can be numbers, variables, or other mathematical objects, while vector spaces contain elements that are vectors, which represent physical quantities or geometric objects.

How are operations performed in an algebra and a vector space?

In an algebra, operations such as addition, subtraction, multiplication, and division are performed on the elements within the algebra. In a vector space, operations such as vector addition, scalar multiplication, and dot or cross products are performed on the vectors within the space.

Can an algebra be considered a vector space?

No, an algebra cannot be considered a vector space because it does not necessarily contain vectors and does not have the same operations as a vector space. However, some algebras may have similarities to vector spaces, such as the algebra of complex numbers which can be viewed as a two-dimensional vector space.

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