# 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 '19 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 '19 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 '19 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... – PossumP Mar 23 '19 at 14:54