Problem
Given a set of vectors \(V = \{v_1, v_2, v_3\}\) in \(\mathbb{R}^3\) with \(v_1 = [1, 0, 0]\), \(v_2 = [1, 1, 0]\), and \(v_3 = [1, 1, 1]\), find an orthonormal basis for this set using the Gram-Schmidt process.
Solution
Step 1 ：Step 1: Normalize \(v_1\) to get \(u_1\), i.e., \(u_1 = \frac{v_1}{||v_1||}\). Here, \(||v_1|| = \sqrt{1^2 + 0^2 + 0^2} = 1\), so \(u_1 = [1, 0, 0]\).
Step 2 ：Step 2: Subtract the projection of \(v_2\) onto \(u_1\) from \(v_2\) to get \(w_2\), and then normalize \(w_2\) to get \(u_2\). Here, \(w_2 = v_2 - (v_2 \cdot u_1)u_1 = [1, 1, 0] - (1)[1, 0, 0] = [0, 1, 0]\), and \(u_2 = \frac{w_2}{||w_2||} = [0, 1, 0]\).
Step 3 ：Step 3: Subtract the projections of \(v_3\) onto \(u_1\) and \(u_2\) from \(v_3\) to get \(w_3\), and then normalize \(w_3\) to get \(u_3\). Here, \(w_3 = v_3 - (v_3 \cdot u_1)u_1 - (v_3 \cdot u_2)u_2 = [1, 1, 1] - (1)[1, 0, 0] - (1)[0, 1, 0] = [0, 0, 1]\), and \(u_3 = \frac{w_3}{||w_3||} = [0, 0, 1]\).
