- #1

rtareen

- 162

- 32

- TL;DR Summary
- We are trying to find the complete solution to the matrix equation Ax = b where A in an m x n matrix and m is not equal to n.

We are trying to find the complete solution to the matrix equation ##A\vec x = \vec b## where A is an m x n matrix and ##\vec b## can be anything except the zero vector. The entire solution is said to be:

##\vec x = \vec x_p + \vec x_n##

where ##\vec x_p## is the solution for a particular ##\vec b## and ##\vec x_n## is the entire nullspace.

I don't understand this. Why is the nullspace included in the solution, when it is defined to be the solution when ##\vec b = \vec 0##? Or else what is this equation really saying?

##\vec x = \vec x_p + \vec x_n##

where ##\vec x_p## is the solution for a particular ##\vec b## and ##\vec x_n## is the entire nullspace.

I don't understand this. Why is the nullspace included in the solution, when it is defined to be the solution when ##\vec b = \vec 0##? Or else what is this equation really saying?