Linear Algebra Approximations

Question 1

What is the equation to estimate a b vector that is outside of the column space of A?

Answer 1

A xhat = p

p is the projection of the b vector onto the column space

Question 2

Equation for xhat

Answer 2

xhat = (A^TA)^-1 A^T b

Question 3

Equation for p (projection of b onto the column space)

Answer 3

p = [A (A^TA)^-1 A^T] b

Question 4

Equation for P (projection matrix)

Answer 4

P = A (A^TA)^-1 A^T

Question 5

Equation for e (error vector)

Answer 5

e = b - Axhat

Question 6

Equation for finding E with partial derivatives

Answer 6

E=||A(xhat)-b||^2

Take partials of E with respect to xhat, set them equal to zero (to minimize E) and solve for the components of the xhat matrix.