Given vectors v1 = [1, 2, 3] and v2 = [4, 5, 6]. Find an orthonormal basis using the Gram-Schmidt process.
Answer
Step 3: Normalize w2 to get the second basis vector: u2 = w2 / ||w2|| = [4 - \( \frac{32}{14} \), 5 - \( \frac{64}{14} \), 6 - \( \frac{96}{14} \)] / sqrt((4 - \( \frac{32}{14} \))^2 + (5 - \( \frac{64}{14} \))^2 + (6 - \( \frac{96}{14} \))^2) = \( \frac{1}{\sqrt{14}} \) [1, 1, 1].
Step 1 ：Step 1: Normalize the first vector to get the first basis vector: u1 = v1 / ||v1|| = [1, 2, 3] / sqrt(1^2 + 2^2 + 3^2) = \( \frac{1}{\sqrt{14}} \) [1, 2, 3]
Step 2 ：Step 2: Subtract the projection of v2 on u1 from v2 to get an orthogonal vector: w2 = v2 - (u1 . v2) u1 = [4, 5, 6] - \( \frac{1}{\sqrt{14}} \) [1, 2, 3] . [4, 5, 6] * \( \frac{1}{\sqrt{14}} \) [1, 2, 3] = [4, 5, 6] - \( \frac{32}{14} \) [1, 2, 3]
Step 3 ：Step 3: Normalize w2 to get the second basis vector: u2 = w2 / ||w2|| = [4 - \( \frac{32}{14} \), 5 - \( \frac{64}{14} \), 6 - \( \frac{96}{14} \)] / sqrt((4 - \( \frac{32}{14} \))^2 + (5 - \( \frac{64}{14} \))^2 + (6 - \( \frac{96}{14} \))^2) = \( \frac{1}{\sqrt{14}} \) [1, 1, 1].
