Linear Algebra 2 Conceptual Questions (Vector spaces)

[Saladsamurai]

Alrighty then

I am working through Axler's LA Done Right. I have 2 questions for now:

1.) He uses some notation that is confusing me. When referring to \mathbf{R} \text{ or }\mathbf{C} as a set it is in BOLD but when he refers to a vector space as being the set V, it is not in boldface. Why?
2.) I am also a little confused by part of the definition of a Vector Space.

Def: A vector space is a set V along with an addition on V and a scalar multiplication on V such
that the following properties hold: commutativity, associativity, ...etc.

(Here is where I get confused)

The scalar multiplication in a vector space depends upon F (where F can refer to either \mathbf{R}\text{ or }\mathbf{C}). Thus when we need to be precise, we will say that V is a vector space over F instead of simply saying that V is a vector space. For example, Rⁿ is a vector space over R and Cⁿ is a vector space over C.

Can someone clarify what he means by a vector space over F or over R and how that is different from just saying a vector space?

He never actually says what this means.

Thanks!

 

[dx]

A vector space over F means that the 'scalars' are elements of F. So a vector space over R means that the scalars are real numbers. You can also have a complex vector space, where the scalars are complex numbers. In general, F can be any field, which is type of set, of which the real and complex numbers the most familiar examples.

 

[Saladsamurai]

A vector space over F means that the 'scalars' are elements of F. So a vector space over R means that the scalars are real numbers. You can also have a complex vector space, where the scalars are complex numbers. In general, F can be any field, which is type of set, of which the real and complex numbers the most familiar examples.

So if we were to just say that V is a vector space, does that automatically imply over all of F?


Also: dx, do you know why F, C, and R are in boldface whereas V is not?

 

[dx]

So if we were to just say that V is a vector space, does that automatically imply over all of F?

Sometimes we can just say V is a vector space if it is understood from the context what the field F is. There's no such thing as a vector space "over all of F". You must always say what F is. For example, you may say F = R if you want a real vector space, or F = C if you want a complex vector space.

Also: dx, do you know why F, C, and R are in boldface whereas V is not?

I don't think there's any particular reason.

 

[Office_Shredder]

It's possible the author is using different typesets to separate sets from vector spaces. So a plain set or a field would be boldfaced, whereas a vector space, which is actually a quadruple - a set, a field, and a multiplication and addition function - is not boldfaced. Don't try to look too deeply into the difference between a vector space and the set of elements that make up a vector space since it's just likely to make your brain hurt without gaining any real insight at the moment

 

[HallsofIvy]

It looks to me like the author is using bold face for fields as distinguished from the vector spaces.