Question

Determine the dual basis
A={(1,0,0),(1,1,0),(1,1,1)}=(a1,a2,a3)

my answer look different to that one in the book but i don't know where did i made the mistake.

Answer 1

my answer is [1-1 0] ,[1 0 -1 ] and [1 0 0]
but they are saying the answer is
[1 -1 0],[ 0 1 -1] and [ 0 0 1]

help pls!

Answer 2

@Coolsector help pls

Answer 3

i am a bit rusty on this, but as much as I remmeber, dual base vectors should satisfy this: \[e_{i}e^{j}=\delta _{i}^{j}\]

Answer 4

actually i am confused do thet use natural basis or they are using the given vectors to find the answer

Answer 5

The condition i wrote in my previous message equals to 3 systems of 3 equations in 3 unknowns. Solving each system will give you one vector for the dual base. Here goes the first one which would be for the first dual base vector:
1x+0y+0z= 1
1x+1y+0z=0
1x+1y+1z=0

for second vector:
1x+0y+0z=0
1x+1y+0z=1
1x+1y+1z=0

for third:

1x+0y+0z=0
1x+1y+0z=0
1x+1y+1z=1

solve this 3 systems and you got your vectors

Answer 6

@REMAINDER