Unique identity and inverse question.

[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?

 

[fourier jr]

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.

 

[tacman]

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.