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So I've run into a bit of a notation problem on my coursework.

I have a vector, $\vert A\rangle$ expressed in the orthogonal basis $\vert 1\rangle$ and $\vert 2\rangle$ as \begin{align} \vert A\rangle &= \vert 1\rangle + i \vert 2\rangle \end{align}

I need to convert it into bra form to normalize it. Is the answer $\langle A\vert = \vert 1\rangle - i\vert 2\rangle$ or $\langle A \vert = \langle 1\vert - i\langle 2\vert$?

Basically, I get that to convert between bra and ket form, you take the complex conjugate and change a column vector to a row one, but I need to know if you convert the basis vectors into bra form as well.

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    $\begingroup$ Dual vectors are linear functionals that also form a vector space, so every bra has to be expressed in terms of other bras (linear functionals). And having in mind that the relation between kets and bras are conjugate-linear, if you have $|\phi\rangle = \alpha |1\rangle + \beta |2\rangle$, its corresponding bra is: $\langle \phi | = \alpha^* \langle 1|+\beta^* \langle 2|.$ $\endgroup$
    – Ellie
    Feb 7 '16 at 11:31
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The comment from @Ellie is what you need to know. This shows the steps. $$ |A\rangle = |1\rangle + i|2\rangle $$ $$ |A\rangle = 1|1\rangle + i|2\rangle $$ $$ \langle A| = |A\rangle^*$$ $$ |A\rangle^* = \alpha^* \langle 1 | + \beta^*\langle 2 |$$ $$ |A\rangle^* = 1^* \langle 1 | + i^* \langle 2 | $$ $$ |A\rangle^* = 1 \langle 1 | + -i \langle 2 | $$ $$ |A\rangle^* = 1 \langle 1 | -i \langle 2 | $$ $$ |A\rangle^* = \langle 1 | -i \langle 2 | $$ $$ |A\rangle^* = \langle A | = \langle 1 | -i \langle 2 | $$

Apologies if it's a bit verbose, but it should be clear.

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