2 For a total of 4 points use the GramSchmidt process to fin

2. For a total of 4 points use the Gram-Schmidt process to find an orthonormal basis for the space W Span 1 1 point for correctly stating the Gram-Schmidt algorithm, 2 points for cor- rectly implementing the algorithm and 1 point for correctly normalizing the resulting vectors. ANS: 0 25

Solution

Solution: (Gram-Schmidth Process)

Given W = Span { [4, 0, 3]^T, [-1, 1, 0]^T}. Let v₁ = [4, 0, 3]^T, v₂ = [-1, 1, 0]^T

Let w₁ = v₁ = [4, 0, 3]^T

Next compute

w₂ = v₂ - [<v₂,w₁>/<w₁,w₁>]w₁ = v₂ - [<v₂,v₁>/<v₁,v₁>]v₁

= [-1, 1, 0]^T- (-4/25)[4, 0, 3]^T = [(-1+16/25), 1, 12/25]^T = [(-9/25), 1, 12/25]^T

Thus w₁, w₂ form orthogonal basis for W. Normalize these vectors to obtain an

orthonormal basis {u₁, u₂} of W. We have ||w₁||² = 25, ||w₂||² = 850/650= 34/25.

So u₁ = (1/5)[4, 0, 3]^T, u₂ = (5/ \\sqrt{34})[(-9/25), 1, 12/25]^T= (1/ \\sqrt{850})[-9, 25, 12]^T