We know how the addition and multiplication of complex numbers is employed in defining the operations for vectors in $\complex{m}$ (Definition CVA and Definition CVSM). We can also extend the idea of the conjugate to vectors.
Suppose that $\vect{u}$ is a vector from $\complex{m}$. Then the conjugate of the vector, $\conjugate{\vect{u}}$, is defined by \begin{align*} \vectorentry{\conjugate{\vect{u}}}{i} &=\conjugate{\vectorentry{\vect{u}}{i}} &&\text{$1\leq i\leq m$} \end{align*}
With this definition we can show that the conjugate of a column vector behaves as we would expect with regard to vector addition and scalar multiplication.
Suppose $\vect{x}$ and $\vect{y}$ are two vectors from $\complex{m}$. Then \begin{equation*} \conjugate{\vect{x}+\vect{y}}=\conjugate{\vect{x}}+\conjugate{\vect{y}} \end{equation*}
Suppose $\vect{x}$ is a vector from $\complex{m}$, and $\alpha\in\complexes$ is a scalar. Then \begin{equation*} \conjugate{\alpha\vect{x}}=\conjugate{\alpha}\,\conjugate{\vect{x}} \end{equation*}
These two theorems together tell us how we can “push” complex conjugation through linear combinations.