What will be an element of a complexification of real vector space This question is about Linear Algebra Done Right by Sheldon Axler, Section 9.A, page 276.
Complexification of real vector space $V$ denoted ${V_c}$ 
(where $V$ is a real vector space) equals $V$ x $V$. 
An element of this complexification is an ordered pair $(u,v)$, where $u,v$ are element of $V$ but we can write this as $$u + iv$$
Please what will be an element of $V_c$? or What will be an element of a complexification of a real vector space ?. 
I was thinking if $V {\epsilon}$ $R^3$ and V = $\Bigg(\begin{matrix}x \\ y \\ z \end{matrix}\Bigg)$ then $V$ x $V$ will be elements of the form 
$$ 
a = \Bigg(\begin{matrix}x \\ y \\ z \end{matrix},\begin{matrix}x \\ y \\ z \end{matrix}\Bigg)
$$
Example of what I was thinking is, if $b$ = $\Bigg(\begin{matrix}1 \\ 2 \\ 3 \end{matrix}\Bigg) \epsilon R^3$ and $b$ is multiplying every element of $R^3$ to form some elements in $T_c$ then some of the elements will be of the form
$$ 
c = \Bigg(\begin{matrix}1 \\ 2 \\ 3 \end{matrix},\begin{matrix}x \\ y \\ z \end{matrix}\Bigg)
$$ where $c$ $\epsilon$ $V_c$ and $c$ can be written in the complex form as 
$$\Bigg(\begin{matrix}1 \\ 2 \\ 3 \end{matrix}\Bigg) +
 i\Bigg( \begin{matrix} x \\ y \\ z \end{matrix}\Bigg)$$
where $x,y,z$ $\epsilon$ $R$
Please I want to know the format elements of $V_C$ (complexification of real vector space) will be. 
 A: The elements can be merely thought of as formal sums, the $i$ doesn't mean anything. That is, if you're main concern is what matrices look like, then note that if $\vec{v} \in V$ where $V$ is some $3$-dimensional vector space, then if you were to look at $\vec{v}$ inside $V^\mathbb{C}$, it would look like:
$$\vec{v}+\vec{0}i=\begin{pmatrix}v_1 \\ v_2 \\ v_3\end{pmatrix}+\begin{pmatrix}0 \\ 0 \\ 0\end{pmatrix} i = \begin{pmatrix}v_1+0i \\ v_2+0i \\ v_3+0i\end{pmatrix} = \begin{pmatrix}v_1 \\ v_2 \\ v_3\end{pmatrix}.$$
With $v_1,v_2,v_3 \in \mathbb{R}$.
To talk about an element of $V^\mathbb{C}$, you need two vectors from $V$, as they are all of the form
$$\vec{u}+\vec{w}i = \begin{pmatrix}u_1 \\ u_2 \\ u_3\end{pmatrix} + \begin{pmatrix}w_1 \\ w_2 \\ w_3\end{pmatrix}i = \begin{pmatrix}u_1+w_1i \\ u_2+w_2i \\ u_3+w_3i\end{pmatrix}.$$
We still consider $V^\mathbb{C}$ as a real vector space, but you can consider it as a vector space over $\mathbb{C}$ as well by defining this kind of scalar multiplication: for $\alpha \in \mathbb{C}$, we have
$$\alpha(\vec{u}+\vec{w}i) = (a+bi)(\vec{u}+\vec{w}i) = \begin{pmatrix}(a+bi)(u_1+w_1i) \\ (a+bi)(u_2+w_2i) \\ (a+bi)(u_3+w_3i)\end{pmatrix}$$
and using the rule that $i^2=-1$.
There are much deeper ways to view this space, such as an extension of scalars and through tensor products, but I believe this is what you were looking for.
