Page 126 - 35Linear Algebra

126 Matrices

where

0 1 0 −i 1 0
σ x = , σ y = , σ z = .
1 0 i 0 0 −1
These three matrices are the famous Pauli matrices; they are used to describe electrons
in quantum theory, or qubits in quantum computation. Let

−2 + i 1 + i
v = .
3 − i 2 − i
Find the column vector of v in the basis B.
For this we must solve the equation

−2 + i 1 + i x 0 1 y 0 −i z 1 0
= α + α + α .
3 − i 2 − i 1 0 i 0 0 −1
This gives four equations, i.e. a linear systems problem, for the α’s

 x y
 α − iα = 1 + i

α + iα = 3 − i
 x y
α z = −2 + i


 z
−α = 2 − i
with solution
y
x
z
α = 2 , α = 2 − 2i , α = −2 + i .
Thus
 
2

−2 + i 1 + i
v = =  2 − i  .
3 − i 2 − i
−2 + i
B
To summarize, the column vector of a vector v in an ordered basis B =
(b 1 , b 2 , . . . , b n ),
 1 
α
 α 2 
 .  ,


.
 . 
α n
is defined by solving the linear systems problem
n
X
i
n
1
2
v = α b 1 + α b 2 + · · · + α b n = α b i .
i=1
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