Definition of the real part of a vector in C^n

**Definition:** The *real part* of a vector \(\mathbf{z} = (z_1, z_2, \ldots, z_n) \in \mathbb{C}^n\) is the vector in \(\mathbb{R}^n\) obtained by taking the real part of each component:

\[\text{Re}(\mathbf{z}) = (\text{Re}(z_1), \text{Re}(z_2), \ldots, \text{Re}(z_n))\]

where \(\text{Re}(a + bi) = a\).

**Alternative formula:** \(\text{Re}(\mathbf{z}) = \frac{\mathbf{z} + \overline{\mathbf{z}}}{2}\)

**Examples:**
- \(\mathbf{z} = (1+i, 2-i, 3)\) → \(\text{Re}(\mathbf{z}) = (1, 2, 3)\)
- \(\mathbf{w} = (i, -2+3i, 5)\) → \(\text{Re}(\mathbf{w}) = (0, -2, 5)\)

**Property:** For any \(\mathbf{z} \in \mathbb{C}^n\):

\[\mathbf{z} = \text{Re}(\mathbf{z}) + i \cdot \text{Im}(\mathbf{z})\]

This decomposes any complex vector into its real and imaginary parts.Copy