Solving Matrix Ax = b Problem: Step-by-Step Homework Guide

[bodensee9]

Homework Statement

I am wondering if someone could help me with the following. I am asked to find a 2 by 3 system Ax = b whose complete solution is x = [1 2 0] + w*[1 3 1] Imagine that these are colum matrices because I can't type columns on here.

FIrst, I don't see how you can get a three row matrix if from a 2 by three system? Don't you need a 3 by 3 matrix if your variable is u,v,w since you have three columns?

So, I guess the eventual matrix would looke something like the following:

1 -1 0
0 -3 1
0 0 0

And if the above is A, would be solution be Ax = [1 2 0]?

And then I'm asked to find a 3 by 3 system with these solutions for Ax = b when b = [b1, b2, b3] and b1+b2 = b3? Imagine again that these are column matrices. So would I then get the following matrix:

1 -1 0
3 0 1
0 0 0

so if that is A then I set it to Ax = [1 2 0]. What I did is that I set A from the first part to equal to some b, that satisfies the condition, so say [1 -1 0] and then use matrix addition and substraction to arrive at the new matrix?

Thanks much!

 

[radou]

Homework Statement

I am wondering if someone could help me with the following. I am asked to find a 2 by 3 system Ax = b whose complete solution is x = [1 2 0] + w*[1 3 1] Imagine that these are colum matrices because I can't type columns on here.

What does Gaussian elimination tell you about the system? How does the system look like when you can 'read' the particular and homogenous solution out of it?

Further on, d = n - r(A), where d is the number of 'parameters in the solution', n is the number of unknowns, and r(A) is the rank of the system matrix A. This relation should be of some use, too.

 

[bodensee9]

You need to do elimination until you get pivots and as much as you can zeros above the pivots. So here the parameter is 1 and so you need to have 3 columns and 2 rows. But then, if that is true, then how can your solution have 3 rows since if your have 3 unknowns and A only has 2 rows? Thanks.

 

[radou]

But then, if that is true, then how can your solution have 3 rows since if your have 3 unknowns and A only has 2 rows? Thanks.

Rows don't matter. If you have 3 unknowns, then your solution vector has to have 3 rows.

Edit: more precisely, after doing Gaussian elimination, you'll obtain a matrix of the form
$$\left( \begin{array}{cccc} 1 & 0 & a_{13}' & | b_{1}' \\ 0 & 1 & a_{23}' &| b_{2}' \\ \end{array} \right)$$
Do you see how the solutions fit in now?