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Prove that for three operators A, B, C we have the following relations: [A, B+C]=[A,B]+[A,C] \\...

Question:

Prove that for three operators {eq}A, B, C {/eq} we have the following relations:

{eq}[A, B+C]=[A,B]+[A,C] \\ [A, BC]=B[A, C] +[A,B] C \\ [AB, C] = A[B,C]+[A,C]B {/eq}

Commutators:

Recall that the commutative property from algebra states

{eq}\begin{align*} ab = ba \end{align*} {/eq}

For real numbers this is of course to, and it's easy to take for granted. Unfortunately, not all operations commute (e.g. matrix multiplication). The commutator is designed as the remainder of the difference of these products:

{eq}\begin{align*} [a,b] &= ab - ba \end{align*} {/eq}

Answer and Explanation: 1

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Part A

We have a short series of identities to prove. In each case, we will expand the commutator and simplify. Remember that order matters. First,...

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The Commutative Property: Definition and Examples

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Chapter 1 / Lesson 10
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Learn what the commutative property is. Discover examples of the commutative property of multiplication and addition, and examine uses of the commutative property.


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