# The Complete Solution to the matrix equation Ax = b

• I
• rtareen
In summary, the complete solution to the matrix equation ##A\vec x = \vec b## includes both a particular solution ##\vec x_p## and the entire nullspace ##\vec x_n##, as represented by the equation ##\vec x = \vec x_p + \vec x_n##. This is because the nullspace can always be added to the particular solution without affecting the result, and any vector in the set ##\vec x_p + \operatorname{null}(A)## is a possible solution to the equation.
rtareen
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?

We have for any x:
##Ax= A(x_n+x_p)= b##
Note that ##x_n## can always be added to the solution since ##Ax_n=0## and ##Ax_p=b##

DaveE and rtareen
If we have two particular solutions ##\vec{x}_p## and ##\vec{x}_q## then ##A\vec{x}_p-A\vec{x}_q=\vec{b}-\vec{b}=\vec{0}##. Thus ##\vec{x}_p-\vec{x}_q \in \operatorname{null}(A)##. Now
##A\vec{x}_p =\vec{b}=A\vec{x}_q + A(\vec{x}_p-\vec{x}_q)=A\vec{x}_p+A\vec{x}_n.##

On the other hand, if we start with any ##\vec{x}_n\in \operatorname{null}(A)##, and have one particular solution ##\vec{x}_p,## then ##A(\vec{x}_p+\vec{x}_n)=\vec{b}+\vec{0}=\vec{b}.## So we may add any vector from the nullspace and get another particular solution ##\vec{x}_p+\vec{x}_n.##

This mean that all vectors in the set ##\vec{x}_p +\operatorname{null}(A)## are all possible solutions to ##A\vec{x}=\vec{b}.##

rtareen and Abhishek11235
These answers make sense. Thanks guys.

## What is the matrix equation Ax = b?

The matrix equation Ax = b is a mathematical representation of a system of linear equations, where A is a matrix of coefficients, x is a vector of variables, and b is a vector of constants. The goal is to find a solution for x that satisfies all of the equations in the system.

## What is the complete solution to the matrix equation Ax = b?

The complete solution to the matrix equation Ax = b is a set of values for x that satisfies all of the equations in the system. This solution can be found using various methods, such as Gaussian elimination or matrix inversion.

## Why is the complete solution to the matrix equation Ax = b important?

The complete solution to the matrix equation Ax = b is important because it allows us to solve complex systems of linear equations and find the values of the variables that satisfy them. This is useful in many fields, including physics, engineering, and economics.

## How do you find the complete solution to the matrix equation Ax = b?

The complete solution to the matrix equation Ax = b can be found using different methods, such as Gaussian elimination, matrix inversion, or using software programs like MATLAB. The method used will depend on the size and complexity of the system.

## What are some real-life applications of the complete solution to the matrix equation Ax = b?

The complete solution to the matrix equation Ax = b has many real-life applications, such as solving systems of equations in engineering design, predicting stock market trends in economics, and analyzing data in scientific research. It is also used in computer graphics and image processing to manipulate and transform images.

Replies
27
Views
2K
Replies
5
Views
2K
Replies
1
Views
297
Replies
2
Views
2K
Replies
3
Views
1K
Replies
1
Views
1K
Replies
12
Views
2K
Replies
4
Views
2K
Replies
3
Views
931
Replies
2
Views
2K