About bases of complex (Hilbert) space

[KFC]

Hi there,
In 3-dimensional real linear space, the simplest bases can be taken as the canonical bases

$$\hat{x} = \left(\begin{matrix}1 \\ 0 \\ 0\end{matrix}\right), \qquad \hat{y} = \left(\begin{matrix}0 \\ 1 \\0\end{matrix}\right), \qquad \hat{z} = \left(\begin{matrix}0 \\ 0 \\ 1\end{matrix}\right)$$

I wonder what's the simplest counterpart for 3-dimensional in complex (hilbert) space?

 

[msumm21]

the basis you gave for R^3 is also a basis for C^3

 

[KFC]

Thanks. But can we construct a basis similar to those for R^3 but with the entries be complex? and how?

 

[Ben Niehoff]

The exact same basis will do. Remember that your scalars are complex numbers.

 

[msumm21]

Thanks. But can we construct a basis similar to those for R^3 but with the entries be complex? and how?

sure, if you replace your 1's with i's it's still a basis. all you have to do is pick 3 vectors so that no one of them is a linear combination of the other 2. if you want it to be an orthonormal basis (which is usually useful) then you need to also make sure that each vector is of unit length and orthogonal to the other 2.