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Prove that the unit coordinate vectors form a basis for Cn

The unit coordinate vectors in \mathbb{R}^n are defined by

    \[ E_i = (0,0, \ldots, 0,1,0,\ldots, 0) \]

where the 1 is in the ith position. We know that these form a basis for \mathbb{R}^n. Prove that they also form a basis for \mathbb{C}^n.


Proof. First, E_1, \ldots, E_n span \mathbb{C}^n since if (z_1, \ldots, z_n) \in \mathbb{C}^n is any vector then

    \[ (z_1, \ldots, z_n) = z_1 E_1 + \cdots + z_n E_n. \]

Further, E_1 ,\ldots, E_n are independent in \mathbb{C}^n since

    \begin{align*}  z_1 E_1 + \cdots + z_n E_n &= 0 & \implies && z_1 &= 0 \\  &&&& z_2 &= 0 \\  &&&& &\vdots \\  &&&& z_n &= 0. \end{align*}

Hence, z_1 = z_2 = \cdots = z_n =0. Therefore, the unit coordinate vectors form a basis for \mathbb{C}^n.\qquad \blacksquare

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