• Solving Ax=b

Three cases:

1. A is squared with full rank

can be solved using multiple ways

• Gauss Elimination
• Inversion of A: x = A^-1 b
• Cholesky decomposition
• QR decomposiiton
• conjugate gradients

2. common real-world situation

overdetermined

no exact solution exists

we try to find approximate result

x* = argmin||Ax-b|| —-> ordinary least squares approach

solution can be obtained through

underdetermined

— infinitively many solutions (not enough information)

• Solving Ax=0 using SVD

Ax = 0 (Homogeneous System)

• we are interested in finding non-trivial solution