Linear Algebra - Vector Spaces over Fields

[KindofSlow]

Hello All,

I’m reading Axler “Linear Algebra Done Right”
I’m still very much a beginner in linear algebra.
My other linear algebra sources all only teach, at least so far, using just Real numbers.
Axler uses both Real and Complex numbers, using symbols R, C, and F where F is all of both real and complex numbers, where R and C are “fields”.

I understand scalar multiplication fairly well, as in ra where r is the scalar and a is the vector.
And I understand what this means: f(r(a)) = r(f(a)), where f is a linear operator.
In the discussion about scalar multiplication, Axler says “…V is a vector space over F…”
Also relevant to my questions is the fact that R is a subset of C

Finally, my questions:

1. Are there limitations or requirements with respect to the combinations of the scalar multiplier and the components of the vector being members of R and/or C. In other words, if vector components are in R only or C, can the scalar multiplier be either? For example I’m wondering if the components of the vector are in R only, can the scalars be C, or vice versa, or is either of these not allowed for some reason.

2. If it is in fact allowed that scalar multipliers and vector components can be from different fields R and C, then is V a vector space “over” the field to which the scalar multipliers belong or the field to which the vector components belong?

As always, I hope this makes sense and thank you very much for any insight you can share.

 

[KindofSlow]

Hello Everyone,
I've been given an explanation. The "over" refers to the field in which the scalar multipliers are in. So a vector space over R means the scalar multipliers are always real numbers. The components of the vectors don't need to be in the field. They just need to be components of vectors such that they follow the addition and scalar multiplication rules of vectors, and that's it.
Thanks everyone and apologies to anyone who could not make sense of my questions.

 

[pka]

I’m reading Axler “Linear Algebra Done Right”
My other linear algebra sources all only teach, at least so far, using just Real numbers.
Axler uses both Real and Complex numbers, using symbols R, C, and F where F is all of both real and complex numbers, where R and C are “fields”.

I've been given an explanation. The "over" refers to the field in which the scalar multipliers are in. So a vector space over R means the scalar multipliers are always real numbers. The components of the vectors don't need to be in the field. They just need to be components of vectors such that they follow the addition and scalar multiplication rules of vectors, and that's it.

I am glad that you received some advice that you found helpful. From what you wrote I had concerns about what you were able to understand.
Do you have a solid background in field theory and in Abelian group theory?
Also, although I have never reviewed Axler “Linear Algebra Done Right”, I am always cautious of any text that says "done right".
May I suggest that you look at Larry Smith's textbook Linear Algebra .

 

[KindofSlow]

Hello pka,
Your concerns are totally justified. I have no background in field theory or Abelian group theory.
I am just an old math hobbyist learning for fun from textbooks with no formal education or classes.
My current Linear Algebra library includes Hefferon, Axler, Lay, Anton, and Strang.
I will keep my eye out for Larry Smith's book at the used book stores I frequent and if I see one I'll definitely pick it up.
I'm pretty good at basic calculus. I'm on the last chapter of Anton's Calculus book which I have enjoyed and is very good.
So I'd also like to pick up an "Advanced Calculus" textbook even though I'm not sure what Advanced Calculus is called - but I think it
includes proving the stuff in basic Calculus, like the power rule, etc. Forgive me if I'm wrong.
So if you also have a recommendation for "Advanced Calculus" that comes after basic Calculus, feel free to send along.
Thank you for your reply - much appreciated.

 

[pka]

So I'd also like to pick up an "Advanced Calculus" textbook even though I'm not sure what Advanced Calculus is called - but I think it includes proving the stuff in basic Calculus, like the power rule, etc. Forgive me if I'm wrong.
So if you also have a recommendation for "Advanced Calculus" that comes after basic Calculus

No one knows what exactly Advanced Calculus means. A book that I like very much is MULTIVARIABLE CALCULUS by Smith & Minton. It is a thin volume that includes almost all of what any vector analyis course would. Introduction To Real Analysis, by Bevan Youse is and old book but very readable.
If you can find a good used edition of A First Course in Calculus by Serge Lang, get it for it makes a great reference.
For serious study in analysis try Undergraduate Analysis by Serge Lang .