Unique identity and inverse question.

In summary, when we say that the identity and inverse element in a vector space is unique, it means that for every element x in the vector space, there is only one identity and one inverse element that satisfies the properties of the identity and inverse. This is true for all elements in the vector space. Additionally, a unique identity means that the entire algebraic system, such as a group or field, has only one element with the properties of an identity. Similarly, a unique inverse means that there is only one element with the properties of an inverse for each element in the system. As for a geometric explanation of a field, it can be thought of as a set of numbers that can be used for both addition and multiplication operations. Lastly,
  • #1

loli12

When we say that the identity and inverse element in a vector space is unique, does it mean that those elements are the same for all x in V? or does it mean that each x has its own unique identity or inverse element?

moreover, is there a geometric way of explaning what a field is? because I got confused about the scalar multiplication of a vector space that b in F and c in V which give bc in V..

and also, did anyone used Linear Algebra by Friedberg, Insel and Spense before? What do you think about the book?
 
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  • #2
unique identity means the whole algebraic system (group, ring or field, etc) has exactly one element e with the property that ae = ea = a for EVERY a in the object you're interested in. same goes for 'unique inverse' & so on.
 
  • #3


The uniqueness of the identity and inverse elements in a vector space means that for any given vector x, there is only one identity element and one inverse element that satisfies the necessary conditions. This means that each x has its own unique identity and inverse element, but they are not necessarily the same for all x in the vector space.

To understand the geometric explanation of a field, it is helpful to think of it as a set of numbers that can be used for both addition and multiplication. In a vector space, the field is used to scale the vectors, which can be thought of as stretching or shrinking the vector in a certain direction. So, the scalar multiplication of a vector space is the combination of the field element (b) and the vector (c) to produce a new vector (bc) in the same direction as c, but with a different magnitude determined by b.

I have not personally used the Linear Algebra book by Friedberg, Insel, and Spence, but it is a well-respected textbook in the field. It covers a wide range of topics in linear algebra and has many examples and exercises for practice. Overall, it is a comprehensive and rigorous book for learning linear algebra.
 

What is a unique identity?

A unique identity is a characteristic or set of characteristics that distinguish an individual from others. It is often used in the context of personal identification, such as a name, social security number, or fingerprint.

Why is having a unique identity important?

Having a unique identity is important for a variety of reasons. It allows individuals to be recognized and differentiated from others, which can be crucial for legal and administrative purposes. It also helps maintain privacy and security in personal and financial transactions.

What is an inverse question?

An inverse question is a question that is formed by switching the subject and the verb of a declarative sentence. For example, the inverse question of "She is going to the store" would be "Is she going to the store?"

How are unique identity and inverse question related?

Unique identity and inverse question are related in that they both involve identifying and distinguishing individuals in some way. Inverse questions can be used to clarify or confirm a person's unique identity, while unique identities can be used to answer inverse questions about that person.

What are some examples of unique identities?

Examples of unique identities include names, social security numbers, passport numbers, driver's license numbers, and fingerprint or DNA profiles. These are all characteristics that are specific to an individual and help differentiate them from others.

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