# Thread: Standard Basis of the complex filed?

1. ## Standard Basis of the complex filed?

Hi,

What is the standard basis of the complex field, C^n?
Like C^3 for example. Would it be the same as R^3?

2. Originally Posted by thejinx0r
Hi,

What is the standard basis of the complex field, C^n?
Like C^3 for example. Would it be the same as R^3?
You can use $\bold{i},\bold{j},\bold{k}$ as a basis. Do not get confused, $i\not = \bold{i}$. Those are different things.

3. Thanks for that, but I'm still having some troubles doing this problem. Or I am a little unsure of my answer.

So here's the question.
I attached the image below, because it had all the proper formatting.
So I calculated that
$T=\left(
\begin{array}{cc}
1 & 1 \\
6 & 0
\end{array}
\right)
$

And that the change-of-basis matrix is
$P=\left(
\begin{array}{cc}
i & 0 \\
0 & 1+i
\end{array}
\right)
$

And it follows that
$P^{-1}=\left(
\begin{array}{cc}
-i & 0 \\
0 & \frac{1-i}{2}
\end{array}
\right)
$

The next part is where I am unsure.

So the matrix T with respect to the new basis would just be
$[T]_{U,U}=P^{-1}\cdot T \cdot P$
but this is just equal to T.
Is that possible?

4. Originally Posted by thejinx0r
The next part is where I am unsure.

So the matrix T with respect to the new basis would just be
$[T]_{U,U}=P^{-1}\cdot T \cdot P$
but this is just equal to T.
Is that possible?
I did not actually do any computations for this problem (so it is possible that you make a mistake in your computations) but I wanted to say if they are equal then so what? Why is that a problem? If you follow the correct way of getting your computations then your answer would be correct regardless if the new matrix is identical to the old matrix.