Existence and Uniqueness of a Linear Least Squares Solution

[scotchtapeman]

I'm studying for my numerical analysis final on tuesday, and I know this is going to be one of the problems, so any help is greatly appreciated.

Homework Statement

State and prove existence and uniqueness for the solution of the linear least squares problem.

Homework Equations

$y \approx x B$
$x' x B = x' y$

The Attempt at a Solution

linear least squares finds B such that $\| y - x B \|$ is minimized.

Since this is linear least squares, $y = B_0 + B_1 x$

$r_i = y_i - (B_0 + B_1 x_i)$
For $1 \le i \le n, \delta r_i / \delta x_i = 0$
Then $(\delta y_i / \delta x_i) - B_1 = 0$

I missed this lecture and I can't find much help online, so I could be headed in the wrong direction. Thanks!

 

[lanedance]

so say you want to find a,b such that it minimises the least square error in y = a + bx

start with the sum of the squares, to characterise the error for given a and b, then minimise w.r.t. a,b and you should be most of the way there