Consider the basis #e_1 = (−2,4,−1)#, #e_2 = (−1,3,−1)# and #e_3 = (1,−2,1)# of #R^3# over #R#. Find the dual basis of #{e_1, e_2, e_3}#.?

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Answer 1 · 2018-02-11T22:28:58

See below.

Calling #e^i# the dual basis of #e_j# we have

#<< e^i, e_j >> = delta_j^i# or

#(e^1, e^2, e^3)((e_1),(e_2),(e_3)) = I_3# then if

#M = ((e_1),(e_2),(e_3)) = ((-2,4,-1),(-1,3,-1),(1,-2,1))# then

#M^-1 = (e^1, e^2, e^3) = ((-1, 2, 1),(0, 1, 1),(1, 0, 2))#

or

#e^1 = (-1,0,1)^top#
#e^2=(2,1,0)^top#
#e^3 = (1,1,2)^top#

form the dual basis.