On page 13 of doCarmo's book on Differential geometry of curves and surfaces he stated that:

...we prove the relation:

where all are arbitrary vectors. This can easily be done by observing that both sides are linear in Thus it suffices to check that:

for all

My question: what does it mean when the author said that

" "? What did he mean by the word linear?

Also the author said that:

where he defined ?

Why is it sufficient to prove the above statement of the author for standard basis only?

How does this proof using standard basis automatically validates for non-standard basis? I know the author is right. But isn't it something like choosing a particular basis to prove our theorem which can be false for other non-standard bases?

I can't find the answer. Is it possible to kindly help me detect what I'm lacking in understanding this statement?