Final answer:
Use the Gram-Schmidt process to determine an orthonormal basis for the subspace of R3 spanned by x and y.
Step-by-step explanation:
The Gram-Schmidt process is used to determine an orthonormal basis for a subspace. To apply this process, start with the given vectors x = [-3, -1, -4] and y = [-4, 2, -4].
Step 1: Normalize the first vector:
x1 = x / ||x|| = [-3/6, -1/2, -4/8] = [-1/2, -1/2, -1/2]
Step 2: Project the second vector onto the subspace spanned by the first vector and subtract the projection:
y1 = y - (y · x1) * x1 = [-4, 2, -4] - (-8/4) * [-1/2, -1/2, -1/2] = [-4, 2, -4] + [2, 2, 2] = [-2, 4, -2]
Step 3: Normalize the second vector:
y2 = y1 / ||y1|| = [-2/sqrt(24), 4/sqrt(24), -2/sqrt(24)]
Therefore, the orthonormal basis for the subspace spanned by x and y is {x1, y2} = {[-1/2, -1/2, -1/2], [-2/sqrt(24), 4/sqrt(24), -2/sqrt(24)]}.