Question

The following wave function |A> given as:
|A> = (9-2i) |u_1> + 4i |u_2> - |u_3 + i|u_4

(a) Find the dual vector <A|
(b) If |u_1> is under an orthonormal base, find the representation of |A> as a column vector in the basis given.

Answer 1

a)

$$\vec{A^*}=\frac{1}{A} \begin{pmatrix} 9+2i & -4i & -1 & -i \end{pmatrix}\\\vec{A^*}=\frac{1}{\sqrt {103}} \begin{pmatrix} 9+2i & -4i & -1 & -i \end{pmatrix}$$

b)

$$\vec{A}= \begin{pmatrix} 9-2i \\ 4i \\ -1 \\ i \end{pmatrix}$$