Matrix / Vector bracket - Transposed

Question

I have the following question: H is a positiv definite matrix, $\lambda$ is a number, $c,x$ and $d$ are vectors

why is

$0=(H(x+\lambda d)+c)^Td = d^THx+\lambda d^THd+d^Tc$

I thought: $(A+B)^T = A^T+B^T$

So I have no idea, why the vector $d$ (right of the bracket) become $d^T$

Thanks for your help

Answer 1

\begin{equation} 0=(H(x+\lambda d)+c)^Td \end{equation} Transposing the whole equation bearing in mind the the left hand side is the scalar zero so it won't change after transpose as well as keeping in mind that $(Bz)^T=z^TB^T$ and $(z^T)^T = z$, we get:

\begin{equation} [(H(x+\lambda d)+c)^Td ]^T=d^T(H(x+\lambda d)+c) = \end{equation}

\begin{equation} = d^THx+\lambda d^THd+d^Tc \end{equation}