All bases are equivalent! So prove something in one basis and you have proven it for all possible bases. There is a similarity matrix that can be applied to transform everything from one basis to another.
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?
All bases are equivalent! So prove something in one basis and you have proven it for all possible bases. There is a similarity matrix that can be applied to transform everything from one basis to another.