# Prove: Linear maps between vectors spaces form a vector space

In order to understand the resultant vector spaces better, could someone provide a proof of the following statement for finite vector spaces:

“Linear maps between vector spaces form a vector space.”

I have no problems for everyday vectors (points in space, ordered n-tuples, etc) and for standard vector addition. However, the definitions clearly go way beyond the every day to operations that are not normal addition, and to objects much more abstract than everyday vectors.

• This may help? – Theo Bendit Mar 23 at 0:39
• Honestly, the proof of this was never helpful while I was learning, because it's pretty much definition crunching. – Don Thousand Mar 23 at 0:41
• It's really best if you work this through yourself. There's nothing to it but checking the axioms. Is there a $0$? Sure, the map that takes everything to $0$. Is a constant times a linear operator linear? Sure. Is the sum of linear operators linear? Sure. Just go step by step. – lulu Mar 23 at 0:46
• @Theo, yes that’s getting at at least one of my sources of confusion, namely the potentially very different definitions of “+” in the two vector spaces. Will read carefully and perhaps return with questions... – MPitts Mar 23 at 14:54

## 1 Answer

So you need to understand the "usual" operations on functions. If f and g are functions, their sum, f+ g, is defined as the function defined by f+g= f(x)+ g(x) and scalar multiplication, af, by af= af(x).

(Yes, this is "definition crunching". And it teaches students how important it is to learn and use precise definitions.)

• So, the vector space of the linear operators essentially “inherits” the definition of “+” from the target vector space; right? – MPitts Mar 23 at 15:31