Let T be a linear map from $\mathbb C \times \mathbb C$ to $\mathbb C \times \mathbb C$ defined by $T(z,w)=(z,0)$.
What is the matrix of the transformation . If it is a real vector space,i can easily do it .But complex vector space itself have (for example)${(0,1),(1,0)}$ as a basis but what is basis of complex vector space with higher dimensions?
how can elements of higher dimensional complex vector space as a linear combination of basis?.
Any hint is appreciated
On
$T$ is a projection onto the first coordinate ( which is a complex number) or, if you prefer, is a projection onto a $2$-dimensional subspace of a $4$-dimensional space.
It is reasonable to work with complex numbers.... didn't you write $T$ operates on $\mathbb{C} \times \mathbb{C}?$ The matrix can be found as suggested by Ernie060 ( method) or simply guessed, and is:
\begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix}
The general approach is to write down the equation $T(z,w)=(z,0)$ in matrix form. So for all $z,w \in \mathbb{C}$ $$ \begin{bmatrix}a & b \\ c & d\end{bmatrix} \begin{bmatrix} z \\ w \end{bmatrix} = \begin{bmatrix} z \\0 \end{bmatrix},$$ where $a,b,c,d \in \mathbb{C}$ are the unknowns we must find. A good trick is to use simple values of $z$ and $w$ so that you get easy equations for $a,b,c,d$. For instance, take $z=1$, $w=0$ and also $z=0$, $w=1$.