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This is a question in the Nielsen and Chuang textbook (Exercise 2.2).

Suppose $V$ is a vector space with basis $|0\rangle$ and $|1\rangle$ and $A$ is a linear operator from $V \to V$ such that $A|0\rangle = |1\rangle$ and $A|1\rangle =|0\rangle$. Given a matrix representation for $A$, with respect to the input basis $\{|0\rangle, |1\rangle\}$ and output basis $\{|1\rangle, |0\rangle\}$. Find input and output basis which give rise to a different matrix representation of A.

My doubts:

  1. Why do we mention the input and output specifically? Isn't the matrix-dependent only on the input and output as long as the transformation is in the same vector space?

  2. How to solve the second part of the question?

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  • $\begingroup$ Related: Nielsen & Chuang Exercise 2.2 - “Matrix representations: example” $\endgroup$ – Sanchayan Dutta Feb 7 '19 at 13:04
  • $\begingroup$ "Why do we mention the input and output specifically? Isn't the matrix-dependent only on the input and output as long as the transformation is in the same vector space?" I don't understand, aren't you answering yourself? The matrix representation depends on input/output bases, and the question is asking to verify this by computing the matrix representation in a different input/output basis $\endgroup$ – glS Feb 8 '19 at 11:30
  • $\begingroup$ also, does the linked post answer your question? If not, could you specify what in the answer there you find unclear? $\endgroup$ – glS Feb 8 '19 at 11:30
  • $\begingroup$ Yes thank you it did, sorry I did not see the message $\endgroup$ – Fatima Aug 13 '19 at 8:03