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5. Let o1, 02, 03 be the Pauli spin matrices -(* :). --(: 5), -(; :) 1 01 02 03 1 0 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 V2 1 03 V2 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, a3 E C.

5. Let o1, 02, 03 be the Pauli spin matrices -(* :). --(: 5), -(; :) 1 01 02 03 1 0 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 V2 1 03 V2 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, a3 E C.

Advanced Engineering Mathematics
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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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.
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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