# Nielsen & Chuang Exercise 2.2 - Matrix representations in different input and output basis [duplicate]

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?

• – Sanchayan Dutta Feb 7 '19 at 13:04
• "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 – glS Feb 8 '19 at 11:30
• also, does the linked post answer your question? If not, could you specify what in the answer there you find unclear? – glS Feb 8 '19 at 11:30
• Yes thank you it did, sorry I did not see the message – Fatima Aug 13 '19 at 8:03