Projection and Least Squares
Question: What is the projection a vector x onto a subspace V of , and how do we find this projection?
Note: We will discuss 2 methods this time, and the second of these is the easiest.
Note: This method begs the question… How do we find an orthogonal basis for a subspace?
The Gram Schmidt Process (a great way to create an orthogonal basis)
Assumption: V is a subspace of Rn with basis {v1,…,vm}.
Goal: Find an orthogonal basis for V.
Example: Find an orthogonal basis for span , , . Then use this orthogonal basis to find the projection of x onto V.
1 2 3
1 0 1 1
1 2 3 1, , ,
1 2 0 1
2 1 1 1
x v v v
Example: Find the projection of x onto span , , by using the system of normal equations.
1 2 3
1 0 1 1
1 2 3 1, , ,
1 2 0 1
2 1 1 1
x v v v