Does complex linear transformation require conjugate transpose?

Question

I try to generalize linear transformation to complex vector space. When I use a complex square matrix A to represent some linear transformation (it still means linear transformation right?) in that complex vector space and act on some complex vector v in that space, when I am to compute the resultant vector, am I supposed to do A multiplies v or A_conjugate_transpose multiplies v? I am confused because the complex inner product of two vectors requires one of the vector's conjugate transpose.

Answer 1

When you do linear algebra over $\mathbb C^n$ and all you have to deal with are linear transformations, vectors and matrices, you don't have to introduce conjugates. For instance, if $f\colon\mathbb C^2\longrightarrow\mathbb C^2$ is defined by $f(x,y)=(2ix+3y,ix+(1+i)y)$, then its matrix with respect to the standard basis will be $A=\left[\begin{smallmatrix}2i&3\\i&1+i\end{smallmatrix}\right]$. And, if $(x,y)\in\mathbb C^2$,$$f(x,y)=A.(x,y)=(2ix+3y,ix+(1+i)y).$$