1. Least Squares Approximations
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

2. Least Squares Approximations
Suppose you have an inconsistent linear system (i.e. there are no solutions).
How do you find a vector that comes close to solving the system?
We will use a method called “Least-Squares” Approximation.
We have the following system:
Unfortunately there are no exact solutions.
We will look for a solution of the related system:
In this equation, is the orthogonal projection of onto the column space of A.
In other words, is the vector that A maps to that is “closest” to .
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

3. Least Squares Approximations
Here is an example:
We have the following system:
A least-squares solution will satisfy the equation
This is our least-squares solution
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB