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Find an orthogonal basis for the column space of the matrix to the right. and x₂ to find V₂- Recall that V₂ = x2 X2 *V1 -v₁. Compute the inner products that appear in this formula. First compute X₂ *V₁. V₁ *V₁ X₂ V₁ = (5)(-1)+(-6)(3) + (− 1)(2) + (-5)(1) = - 30 Next compute v₁ •V₁. V₁V₁=(-1)(-1) + (3)(3) + (2)(2)+(1)(1) = 15 -1 01 5 7 4 3-6 2 -1 7 1 -5 -4
Substitute -30 for x₂ V₁, 15 for V₁ V₁, and the vectors x₂ and v₁ into the formula for v2. X2 *V1 V1.₁ V₂=X2- ' -6 -1 -5 5 -5 Simplify this result. 5 +2 -1 -30 3 15 -1 3 N 1 N 1 W 0 3 - 3
This means that V₂ = Use X3 - 1 4-H V₁ and V₂- 2 1 X3 *V2 V2 *V2 V3 = X3 -4 3 0 X3 *V1 V₁ •V₁ w 3 3 - 3 to find V3 using the formula shown below.
Compute the inner products that appear in this formula. First compute X3 V₁. X3 V₁ = (7)(-1) + (4)(3) + (7)(2) + (− 4)(1) = 15 Recall that v₁ V₁ = 15. Compute X3 *V2- X3 V₂ = (7)(3) + (4)(0) + (7)(3) + (− 4)(-3) = 54 Compute V₂ *V₂. V₂ V₂ = (3)(3) + (0)(0) + (3)(3) + (− 3)(-3) = 27
Substitute 15 for X3 *V₁, 15 for v₁ •V₁, 54 for x3 *V₂, 27 for v₂V₂, and the vectors X3, V₁, and v₂ into the formula for V3. 1 V3 = x3 || X3 *V1 V₁ *V₁ 7 15 7 15 4 Simplify this result. 2 1 X3 *V₂ V2 *V2 -1 3 2 1 - 3 54 27 3 0 3 - 3
This means that = V3 N and so an orthogonal basis for the column space of the given matrix is V T 3 N m 3 3 N - 1