The Exercise

Take a complex vector space of dimension two with basis vectors $|0\rangle$ and $|1\rangle$, and let $A$ be the linear operator on it that exchanges the two basis vectors.^✦ The exercise asks for the matrix of $A$ with respect to the basis $|0\rangle, |1\rangle$, used for input and output alike, and then for a basis in which $A$ has a different matrix.

The Space and Two Bases

The library works on $\mathbb{C}^2$. The computational basis $\mathcal{C}$ of $\mathbb{C}^2$ is its standard basis, the two unit column vectors $$|0\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \qquad |1\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}.$$(1)^✦

The basis $\mathcal{B}_\pm$ of $\mathbb{C}^2$ consists of the sum and the difference of the computational basis vectors, in this order: $$\mathcal{B}_\pm = \bigl( |0\rangle + |1\rangle,\; |0\rangle - |1\rangle \bigr).$$(2)^✦ Its vectors are not normalized.

The Operator and Its Matrix

The operator $A$ of the exercise is the linear operator on $\mathbb{C}^2$ defined on the computational basis by $$A|0\rangle = |1\rangle, \qquad A|1\rangle = |0\rangle.$$(3)^✦

The matrix $[T]_{\mathcal{B}}$ of a linear operator $T$ on $\mathbb{C}^2$ with respect to a basis $\mathcal{B} = (|b_0\rangle, |b_1\rangle)$, taken for both input and output, is the $2 \times 2$ matrix whose entries satisfy $$T|b_j\rangle = \sum_{i=0}^{1} \bigl([T]_{\mathcal{B}}\bigr)_{ij} \, |b_i\rangle .$$(4)

The Claim

What Is Checked

The claim is stated over the operator $A$ of (3) and over two bases of $\mathbb{C}^2$, the computational basis (1) and the basis $\mathcal{B}_\pm$ of (2).

About these constructions the library proves that $|0\rangle + |1\rangle$ and $|0\rangle - |1\rangle$ are linearly independent.^✦