The Exercise

Take the vectors $|w\rangle = (1,1)$ and $|v\rangle = (1,-1)$ of $\mathbb{C}^2$, given by their components. The exercise asks to verify that they are orthogonal and to find their normalized forms, each vector divided by its norm.

The Vectors

All vectors here lie in $\mathbb{C}^2$ with its standard inner product, $\langle a|b\rangle = a_1^* b_1 + a_2^* b_2$ for $|a\rangle$ and $|b\rangle$ with components $a_i$ and $b_i$, and the norm $\| |a\rangle \| = \sqrt{\langle a|a\rangle}$ it induces.

The vector $|w\rangle$ is the vector of $\mathbb{C}^2$ with both components equal to $1$, $$|w\rangle = \begin{pmatrix} 1 \\ 1 \end{pmatrix}.$$(1)^✦ The vector $|v\rangle$ is the vector of $\mathbb{C}^2$ with components $1$ and $-1$, $$|v\rangle = \begin{pmatrix} 1 \\ -1 \end{pmatrix}.$$(2)^✦

The vector $|\hat w\rangle$ of $\mathbb{C}^2$ has both components equal to $1/\sqrt{2}$, and the vector $|\hat v\rangle$ of $\mathbb{C}^2$ has components $1/\sqrt{2}$ and $-1/\sqrt{2}$: $$|\hat w\rangle = \begin{pmatrix} 1/\sqrt{2} \\ 1/\sqrt{2} \end{pmatrix},$$(3) $$|\hat v\rangle = \begin{pmatrix} 1/\sqrt{2} \\ -1/\sqrt{2} \end{pmatrix}.$$(4)

The Claim

What Is Checked

The theorem is stated over the vectors $|w\rangle$ and $|v\rangle$ of (1) and (2), the vectors $|\hat w\rangle$ and $|\hat v\rangle$ of (3) and (4), and the inner product and norm of $\mathbb{C}^2$.