Fill in imaginary part definition

# Definition of the imaginary part of a vector in C^n

Put content here**Definition:** The *imaginary 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 imaginary part of each component:
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\[\text{Im}(\mathbf{z}) = (\text{Im}(z_1), \text{Im}(z_2), \ldots, \text{Im}(z_n))\]
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where \(\text{Im}(a + bi) = b\) (the imaginary part is a **real** number, not \(bi\)).
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**Alternative formula:** \(\text{Im}(\mathbf{z}) = \frac{\mathbf{z} - \overline{\mathbf{z}}}{2i}\)
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**Examples:**
- \(\mathbf{z} = (1+i, 2-i, 3)\) → \(\text{Im}(\mathbf{z}) = (1, -1, 0)\)
- \(\mathbf{w} = (i, -2+3i, 5)\) → \(\text{Im}(\mathbf{w}) = (1, 3, 0)\)
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**Decomposition:** Any \(\mathbf{z} \in \mathbb{C}^n\) can be written as:
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\[\mathbf{z} = \text{Re}(\mathbf{z}) + i \cdot \text{Im}(\mathbf{z})\]
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Note that \(\text{Re}(\mathbf{z}), \text{Im}(\mathbf{z}) \in \mathbb{R}^n\).

# Parents

* Algebraic properties of R^n (or C^n)