Let $V=C^2$ and $\alpha= (x_1,x_2)$ and $\beta= (y_1,y_2)$ two elements of $V$. Let $g$ the form on $V$ defined by $$g(\alpha,\beta)=2x_{1}\overline{y}_{1}-ix_{2}\overline{y}_1+ix_{1}\overline{y}_{2}+x_{2}\overline{y}_{2}.$$

(a) Show that $g$ is an inner product on $V$.

(b) Find an orthogonal basis inner product space $V$.

(c) If $g$ is an operator on $V$ given by $T (x_{1}, x_ {2}) = (-ix_2, x_1)$ get $T ^ {*}$.

I tried this:

(a) I have to check the properties of the inner product space (4 properties)

(b) Can I use the canonic base?

(c) I have to prove that $\langle T\alpha,\beta\rangle = \langle\alpha,T^*\beta\rangle$ (I did this).

Thank for you attention and your help.

  • $\begingroup$ yes, you can use the canonical base ^^ $\endgroup$ – math student Feb 22 '13 at 1:54
  • $\begingroup$ It might help if for some of the 4 properties you used the rectangular form of complex numbers, and for some others you might use the polar form of complex numbers. $\endgroup$ – Barbara Osofsky Feb 22 '13 at 3:35
  • 1
    $\begingroup$ You wrote down what you have to do and whether you can use the canonical base...now, what have you actually done? $\endgroup$ – DonAntonio Feb 23 '13 at 1:45

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