Word Count : 937

---

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 + ___ ___ ________ __________ ___ ________ ________ _______.
_______ ____ __________ __________ ____ ___ ________ ____.
__________ ____ ________ _____ ________.
______ ______ ______ ________ ______ _______ ___ _________ _____.
_______ ________ ____ _______ ______.
____ ______ _____ __________ _________.
________ _____ __________ ________ _________ __________ ______ _________ ______.
______ ______ _______ ______ __________ _____ __________ _______ _______ ________.
____ ________ _________ __________ ______ ________ _________.
________ ____ ____ ______ _______ ____.
________ _________ _______ ___ _____.
_________ _________ ____ ___ _________ _______ _____ ___.
________ ____ _____ ____ _____.
_________ ______ ________ ____ _________ __________.
______ ________ _____ ______ ____ ________.
__________ _______ _________ ____ _______ ________ ____.
________ ___ _____ ______ _______ __________ ______ __________ _________ _________ ________ ________.
____ ___ _____ _________ ______ ______.
_______ _____ _________ _____ _______.
_________ _____ ______ _____ _____ _______ ______ _______ ___.
_____ ____ ______ _________ ____ _____ ______ ___ __________ ______ ___ _____.
_______ _________ _________ _____ ___ ________.
____ _____ ____ ______ ____ ______ _______ ______.
_________ ____ __________ _______ _______ _____ __________ ____ _____.
___ ____ __________ ____ __________ ________ __________ _________ _______.
___ ___ _____ ___ ___ _______ ___ ___ _____ ____ _______ ___.
________ ____ ____ ____ ___.
________ __________ ________ ___ ______ ___.
________ ______ _____ _________ ____ _______ ______ _______ _______.
___ ___ ______ _________ ___ ____ __________ ____ _________ ____.
____ ____ _____ _____ ______.
_____ __________ _________ _______ ___ _____ ____ ________.
________ _________ ___ _____ ________ _______ ___ ___ _______ ____.
__________ _______ ________ _________ __________ ____ ____ _________ ____ ______ _______.
___ _________ ________ ____ ___ _______.
________ ___ ________ ______ ______ _____ __________ ______.
_______ ____ ____ ___ ___ _____ __________.
__________ _______ _______ ___ ___ ______.
________ _______ ___ ______ ______ ________ __________ _________ _______.
_____ ______ _________ _____ _______ _____ ______.
____ _______ ________ _______ ________ _______ ____ ________ _____ ___.
___ ______ _________ ______ ______ _______ ________ ________ _______ _______.
__________ _____ ________ ____ __________ _________ __________ _________ _______ ___ ____.
_______ ________ ___ _______ _____ _____ ___ _______ ___ _______.
___ ____ _____ __________ _________.
__________ ________ ________ _________ ______ _________ ________ _______.
_____ ______ ___ ___ ______ ________ ______ ___ _____ __________ _______ ____.
___ ____ ______ ____ __________ ____ ________ ____ ________ ______ _________.
__________ _______ ___ _________ __________ ___ __________.
______ _____ ___ _________ __________ _________ ______ ________ ______ _________.
_____ ______ ______ _____ ______ ______ _____ ________ ____ ______.
__________ _______ ___ _____ ___ ___ ______ _________.
___ _____ _________ _________ _______ _______ ___ ___ __________.
______ _________ _________ _________ _________ _________ ___ ____ ________.
__________ ________ ____ ________ __________ _____ _______ ______ ______ __________.
__________ __________ ____ ________ ___ ________.
_______ ________ _________ ______ _____ _________ _____.
_____ _______ ___ ______ _____ __________ ______ __________ ________ ___ ___ ______.
_________ __________ _______ __________ _____ ______ _____ ____.
________ ______ __________ _________ ____ ___ ______ ______ ___.
________ ____ ____ ______ __________ ________ ______ ______ ____ _______ ______ ________.
_____ __________ _________ ____ __________ _________ _______.
___ _______ ______ _________ ______.
__________ _________ __________ ___ ___ ___.
______ __________ _______ _________ _________ _________ _____ _________ _________ ____ _____.
______ _________ _________ _____ ___ ______ ___ _____ _____ _________.
____ _______ ______ _____ _____ _______ _________ ________ _______ ___ _________ _____.
_____ ____ ______ _________ _____ ___ ____ ______ ___ __________ __________.
_____ ________ ____ ___ ________ ___ ________ ___.
_________ ___ ________ _______ ________.
________ ______ _______ ________ _______ ___ __________ ____ ___ ____ ___ ____.
_______ _________ __________ __________ ___ ________ _______ _______ _______ ____ _____.
_______ ________ ______ __________ _________ __________ __________ _________ ____ ________ _______ _________.
_______ _________ _______ _____ ________ _____ __________ _____ _____ ___ _______ ___.
__________ _____ _______ ____ ____.
______ ____ ____ _____ ________ ____.
_________ ___ ___ _________ _______ ___ _______ _______.
____ _______ ________ ____ ______ _________.
__________ ____ _____ ______ _____ _________ ____ _______ __________ ______ _______.
______ _______ ___ ______ __________ ______ _____.
_________ ___ ________ ______ ________ _________ _____ ____ ___ __________ ________.
___ _____ ___ _________ ___ __________.
________ ___ __________ ________ ____ _____ _____ _________ __________ ________ _____ _______.
_________ ______ _________ ___ ______ _____ ____ ____ ________.
________ ____ ______ ________ ____ _______.
_____ __________ ___ ____ ______ __________ ________ ____ _________.
___ __________ ___ _________ _______ ___.
_____ ________ ______ __________ __________.
________ ___ __________ _________ ________.
_______ __________ ___ __________ ___ _______ __________ __________ _________.
______ ________ ________ ____ _____ _____ ______ ____ __________.
__________ _________ ___ ___ _________ __________ _________ ___.
___ ________ ____ _______ _____ ______ ________ __________ _________ _________ _______ ______.
_________ _____ ____ __________ ___ ____ ___ _________ _________ _____.
_____.
Get Full Answer on WhatsApp