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, 2], b = [8, 2, 4]

       [2, -3]

       [-1, 3]

(a) Construct the normal equations for x.

[        ] [    ]   = [ ]

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

(b) Solve for x.

x = [    ] (Simplify your answers.)

      [    ]

Answer 1

Given, A = and b =

a) The normal equations for X are, AX = b

i.e., =

b) Here we use LU decomposition to solve this normal equations.

Then, A = =

i.e., A = LU , where L = and U =

Now we are going to solve LY = b, where Y = .

Then, =

i.e., a = 8

-2a+b = 2

a+b+c = 4

i.e., a = 8, b = 18, c = -22

Therefore, Y =

Now we solve UX = Y

i.e., =

i.e., -x+2y = 8

y = 18

i.e., x = 28, y = 18

Hence, the solution is .

I think there is some wrong information or mistakes in this sum.