316 CHAPTER21. QUANTUMMECHANICSASLINEARALGEBRA
Sinceboth|v′>andM|ei>areketvectors,theymusthaveanexpansioninbasis
vectorswhichwewriteas
|v′>=
∑
k
v′k|ek> (21.77)
and
M|ej>=
∑
k
mkj|ek> (21.78)
Substitutinginto(21.76)gives
∑
k
v′k|ek>=
∑
i
∑
k
vimki|ek> (21.79)
Takingtheinnerproductofbothsideswiththebravector<ei|,andusingagainthe
bi-linearitypropertyandorthonormalityproperties(21.62),(21.63)
<ei|
∑
k
v′k|ek> = <ei|
∑
j
∑
k
vjmkj|ek>
∑
k
vk′ <ei|ek> =
∑
j
∑
k
vjmkj<ei|ek>
∑
k
vk′δik =
∑
j
∑
k
vjmkjδik
v′i =
∑
j
mijvj (21.80)
Butthisisjusttherule,incomponents,formultiplyingacolumnvectorbyamatrix.
Thismeansthat,inagivenbasis,theactionofalinearoperatoronvectorsisequiv-
alenttomatrixmultiplication. Infact,takingtheinnerproductofbothsidesofeq.
(21.78)withthebravector<ei|,wefind
TheMatrixElementofOperatorM inthebasis{|en>}
mij=<ei|M|ej > (21.81)
Next,itis useful to introduce alinearoperationLon brasandkets whichis
representedsymbollicallyby
L=|u><v| (21.82)
ThemeaningofthissymbolisthatLoperatingonany ket|w >willturnitinto
anotherket,proportionalto|u>,bytakinganinnerproductontheleft
L|w> = (|u><v|)|w>=|u>(<v|w>)
= (<v|w>)|u> (21.83)