How do you know that Ax=b can be solved for all b?

[Cade]

Homework Statement

Suppose you know that the 3 by 4 matrix A has the vector S = (2,3,1,0) as the only special solution to Ax = 0.
(a) What is the rank of A and the complete solution to Ax=0
(b) What is the exact row reduced echelon form R of A?
(c) How do you know that Ax=b can be solved for all b?

Homework Equations

r(A) = n - r

The Attempt at a Solution

a) r(A) = n - r(nullspace) = 4 - 1 = 3
x = a(2,3,1,0), a = any scalar
b) R has 3 pivots and Rx=0 with x = (2,3,1,0)
R = (
(1, 0, -2, 0),
(0, 1, -3, 0),
(0, 0, 0, 1))
c) Clueless. :( How should I attempt this?

 

[CompuChip]

OK, what I posted here first was complete nonsense, so let me try again...

How would you normally like to solve Ax = b?

 

[Cade]

I don't know how to find the determinant of R, it's not a square matrix, it's 3x4.

Edit: Find a particular solution for Axp=b by setting free variables to 0, then find special solutions for Ax0 = 0 and express x as xp + x0.

 

[sutupidmath]

A has to be non-singular if the vector equation Ax=b, is to have a unique solution for every column vector b.

 

[Mark44]

A has to be non-singular if the vector equation Ax=b, is to have a unique solution for every column vector b.

But since A is 3X4, there's no chance of A being nonsingular.

However, since dim(ker(A)) = 1, it should be easy to show that the transformation represented by A is onto R³. Hence for any vector b in R³, there is some vector x in R⁴ such that Ax = b.

Also, since Cade has the RREF form of the matrix, he/she can set up an augmented matrix that represents Ax = b (where b is an arbitrary vector in R³), and show b has a preimage vector x.

 

[Cade]

However, since dim(ker(A)) = 1, it should be easy to show that the transformation represented by A is onto R³. Hence for any vector b in R³, there is some vector x in R⁴ such that Ax = b.

Also, since Cade has the RREF form of the matrix, he/she can set up an augmented matrix that represents Ax = b (where b is an arbitrary vector in R³), and show b has a preimage vector x.

I am sorry for the late reply, my homework was just handed back to me today. I tried the approach second and it worked, now I also understand the first. Thanks for your help.