Let U be the subspace spanned by the orthogonal set {[1, 2, 3],[-1, -1,1]} and x=[0, 2, 1]. Let v = (a, b, c) be the projection of x on U. Find a.Select one:a. 5/6b. -2/3c. 5/3d. -7/6e. None of the other choices is correct

The projection of a vector x onto a subspace U spanned by an orthogonal set of vectors is given by the sum of the projections of x onto each vector in the set.

The projection of x onto a vector v is given by the formula:

proj_v(x) = ((x . v) / ||v||^2) * v

where "." denotes the dot product, and ||v|| denotes the magnitude of v.

Given the orthogonal set {[1, 2, 3],[-1, -1,1]} and x=[0, 2, 1], we can find the projection of x onto U by finding the projections of x onto each vector in the set and adding them together.

First, let's find the projection of x onto [1, 2, 3]:

proj_1,2,3 = (([0, 2, 1] . [1, 2, 3]) / ||[1, 2, 3]||^2) * [1, 2, 3] = ((01 + 22 + 1*3) / (1^2 + 2^2 + 3^2)) * [1, 2, 3] = (7 / 14) * [1, 2, 3] = [1/2, 1, 3/2]

Next, let's find the projection of x onto [-1, -1, 1]:

proj_-1,-1,1 = (([0, 2, 1] . [-1, -1, 1]) / ||[-1, -1, 1]||^2) * [-1, -1, 1] = ((0*-1 + 2*-1 + 1*1) / (-1^2 + -1^2 + 1^2)) * [-1, -1, 1] = (-1 / 3) * [-1, -1, 1] = [1/3, 1/3, -1/3]

Adding these two vectors together gives us the projection of x onto U:

v = [1/2, 1, 3/2] + [1/3, 1/3, -1/3] = [5/6, 4/3, 7/6]

So, a = 5/6. The correct answer is (a) 5/6.