5. Let o1, 02, 03 be the Pauli spin matrices - (? ). 0 1 1 0 -(? ) 이-(69) 1 01 = 02 = i 03 and oo = I2. Consider the vector space of 2 x 2 matrices over C. (i) Show that the 2 x 2 matrices 1 1 12, 1 021 are linearly independent. (ii) Show that any 2 x 2 complex matrix has a unique representation of the form aoI2 + ia101 + ia202 + iaz03 for some ao, a1, a2, az E C.
5. Let o1, 02, 03 be the Pauli spin matrices - (? ). 0 1 1 0 -(? ) 이-(69) 1 01 = 02 = i 03 and oo = I2. Consider the vector space of 2 x 2 matrices over C. (i) Show that the 2 x 2 matrices 1 1 12, 1 021 are linearly independent. (ii) Show that any 2 x 2 complex matrix has a unique representation of the form aoI2 + ia101 + ia202 + iaz03 for some ao, a1, a2, az E C.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:5. Let o1, 02, 03 be the Pauli spin matrices
n-(: ¿). n-(: 3). -(; :)
0 1
1 0
1
03 =
01
i
and oo = I2. Consider the vector space of 2 x 2 matrices over C.
(i) Show that the 2 x 2 matrices
1
1
I2,
1
03
1
are linearly independent.
(ii) Show that any 2 x 2 complex matrix has a unique representation of the form
aoI2 + ia101 + ia202 + ia303
for some ao, a1, a2, a3 € C.
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