Question

Linear Algebra

Find a least-squares solution of Ax = b by (a) constructing the normal equations for x and (b) solving for x.

A = [1, -3], b = [5]

      [-1,  3]         [1]

      [0,   2]         [-3]

      [3,   6]         [4]

a. Construct the normal equations for x without solving.

[            ] x = [    ]   (Simplify your answers.)

[            ]       [     ]

b. Solve for x.

x =    (Simplify your answer.)

Answer 1

Given, A = and b = .

a) The normal equations for X without solving : = .

b) We apply LU factorization on the matrix A.

Then, =

i.e., A = PLU, where L = , U = , P =

Now, (PLU)X = b

i.e., P(LUX) = b

First we solve PZ = b, where Z =

i.e., =

i.e., k = 5

m = 1

l = -3

n = 4

Therefore, Z =

Now we solve LY = Z, where Y =

i.e., =

i.e., p = 5

-p+q = -3

r = 1

3p+(15/2)r+s = 4

i.e., p = 5

q = 2

r = 1

s = -37/2

Therefore, Y =

Now, we solve UX = Y

i.e., =

i.e., x-3y = 5

2y = 2

i.e., x = 8

y = 1

Hence, the solution is .