Answer:
To find an orthonormal basis for W = Span{X1, X2, X3}, we can use the Gram-Schmidt process. This involves taking the first vector and normalizing it to obtain the first basis vector, and then subtracting the projection of the second vector onto the first basis vector from the second vector to obtain the second basis vector, and so on.
First, we normalize the first vector X1:
v1 = X1 / ||X1|| = [1/3, 0, 2/3, -1/3]
where ||X1|| is the norm of X1.
Next, we compute the projection of X2 onto v1, and subtract it from X2:
proj_v1(X2) = (X2 · v1) * v1 = [(2/3) / (1/3)] * v1 = [2, 0, 4/3, -2/3]
v2 = X2 - proj_v1(X2) = [-5/3, 1, -4/3, 4/3]
where · denotes the dot product.
Then, we compute the projection of X3 onto v1 and v2, and subtract these from X3:
proj_v1(X3) = (X3 · v1) * v1 = [(2/3) / (1/3)] * v1 = [2, 0, 4/3, -2/3]
proj_v2(X3) = (X3 · v2) * v2 = [-1/3, 2/3, -1/3, 1/3]
v3 = X3 - proj_v1(X3) - proj_v2(X3) = [-1/3, -2/3, 2/3, -1/3]
Finally, we normalize v2 and v3 to obtain the orthonormal basis vectors:
u2 = v2 / ||v2|| = [-sqrt(5)/5, sqrt(5)/5, -2/sqrt(5), 2/sqrt(5)]
u3 = v3 / ||v3|| = [-1/3sqrt(2), -2/3sqrt(2), sqrt(2)/3, -1/3sqrt(2)]
Therefore, an orthonormal basis for W = Span{X
Step-by-step explanation: