# Matrix of complex linear transformation!

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

• It is very helpful if you suggest a general approach or reference rather than answer to the problem at the beginning of the question! THANKS! Commented Sep 23, 2018 at 17:31
• Is $T$ from $\mathbb{C}\times\mathbb{C}$ to $\mathbb{C}\times\mathbb{C}?$ Commented Sep 23, 2018 at 18:27
• @Maam sorry , you are right ... i will edit it Commented Sep 25, 2018 at 16:27

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 $$z=0$$, $$w=1$$.
$$T$$ 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:
$$\begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix}$$