Word Count : 937

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Let the vectors be [ v_1 = (1, i, 0, 1), \quad v_2 = (1, 0, i, 0), \quad v_3 = (-i, 0, 1, -1) ] We want an orthonormal basis using the Gram-Schmidt process in (\mathbb{C}^4). The inner product in (\mathbb{C}^4) is (\langle x, y \rangle = \sum_{k=1}^4 x_k \overline{y_k}). --- Step 1: Set (u_1 = v_1) [ u_1 = v_1 = (1, i, 0, 1) ] Compute its norm: [ |u_1| = \sqrt{\langle u_1, u_1 \rangle} = \sqrt{1\cdot 1 + i\cdot(-i) + 0 + 1\cdot 1} = \sqrt{1 + 1 + 0 + 1} = \sqrt{3} ] So the first orthonormal vector is [ e_1 = \frac{u_1}{|u_1|} = \frac{1}{\sqrt{3}}(1, i, 0, 1) ] --- Step 2: Orthogonalize (v_2) against (u_1) [ u_2 = v_2 - \frac{\langle v_2, u_1 \rangle}{\langle u_1, u_1 \rangle} u_1 ] Compute (\langle v_2, u_1 \rangle): [ \langle v_2, u_1 \rangle = 1\cdot 1 + 0\cdot (-i) + i\cdot 0 + _______ ________ ____ _____ ________ __________ __________ ________ _____.
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