The problem is,
Suppose $V$ is a vector space with basis vectors $|0\rangle$ and $|1\rangle$, and $A$ is a linear operator from $V$ to $V$ such that $A|0\rangle = |1\rangle$ and $A|1\rangle = |0\rangle$. Give a matrix representation for $A$, with respect to the input basis $|0\rangle, > |1\rangle$, and the output basis $|0\rangle, |1\rangle$. Find input and output bases which give rise to a different matrix representation of $A$.
By using the equation $A|v_j\rangle = \sum \limits_{i} A_{ij}|w_i\rangle$ (2.12), I got the matrix representation $A = \begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}$. But I could't understand the latter part,
Find input and output bases which give rise to a different matrix representation of $A$.
How to solve this?
What about another basis say $|1 \rangle - |0 \rangle$, and $|1 \rangle + |0 \rangle$, then $A$ would be different. If I understand correctly, it's basically asking for two different basis vectors of your vector space, and then a linear transformation that takes one basis vector to the other, and vice versa.