Let x=[1, 0, -1], and let U be the subspace spanned by the orthogonal set {[2, 1, -1],[1,1,3]}. Vector x can be written as x=x1+x2, where x1 is in U and x2 is orthogonal to U. Find the first coordinate of x2.Select one:a. 6/11b. 3/11c. None of the other choices is correctd. 4/11e. 5/11

To find the first coordinate of x2, we first need to find the projection of x onto U. The projection of x onto U is given by the formula:

proj_U(x) = ((x . u1)/(u1 . u1)) * u1 + ((x . u2)/(u2 . u2)) * u2

where u1 and u2 are the vectors that span U, and "." denotes the dot product.

Given that u1 = [2, 1, -1] and u2 = [1, 1, 3], and x = [1, 0, -1], we can calculate:

proj_U(x) = (([1, 0, -1] . [2, 1, -1])/([2, 1, -1] . [2, 1, -1])) * [2, 1, -1] + (([1, 0, -1] . [1, 1, 3])/([1, 1, 3] . [1, 1, 3])) * [1, 1, 3]

Calculating the dot products, we get:

proj_U(x) = ((12 + 01 + -1*-1)/(22 + 11 + -1*-1)) * [2, 1, -1] + ((11 + 01 + -13)/(11 + 11 + 33)) * [1, 1, 3]

Simplifying, we get:

proj_U(x) = ((2 + 1)/6) * [2, 1, -1] + ((1 - 3)/11) * [1, 1, 3]

proj_U(x) = [1/2, 1/4, -1/2] + [-2/11, -2/11, -6/11]

proj_U(x) = [6/22, 11/44, -22/44] + [-4/22, -4/22, -12/22]

proj_U(x) = [2/22, 7/44, -10/44]

So, x1 = proj_U(x) = [2/22, 7/44, -10/44]

Then, x2 = x - x1 = [1, 0, -1] - [2/22, 7/44, -10/44] = [20/22, -7/44, 34/44] = [10/11, -7/44, 17/22]

So, the first coordinate of x2 is 10/11, which is not one of the given options. Therefore, the correct answer is:

c. None of the other choices is correct.