Finding basis for a space

Question

I apologize for my easy question but this is new to me My question is why another basis for $T_{a}^{(2)}$ is the phrase that I denote it in the picture?

Answer 1

It's just to mention that in addition to $\mathrm d_ax\wedge\mathrm d_ay$ expressed in $x$ and $y$ being a basis you also have a basis expressed in $z$ and $\overline z$, namely $\mathrm d_az\wedge\mathrm d_a\overline z$ and they just differ by a factor of $-2i$.

This is the result of an easy calculation: \begin{align*} \mathrm d_az\wedge \mathrm d_a\overline z &= \mathrm d_a(x+iy) \wedge \mathrm d_a(x-iy) \\&= (\mathrm d_ax+i\,\mathrm d_ay) \wedge (\mathrm d_ax-i\,\mathrm d_ay) \\&= \underbrace{\mathrm d_ax\wedge\mathrm d_ax}_{0} -i\,\mathrm d_ax\wedge\mathrm d_ay+i\,\underbrace{\mathrm d_ay\wedge\mathrm d_ax}_{-(\mathrm d_ax\wedge\mathrm d_ay)}-i^2\,\underbrace{\mathrm d_ay\wedge\mathrm d_ay}_{0} \\&= -2i\,\mathrm d_ax\wedge\mathrm d_ay. \end{align*}