Absolutely stumped on this one (all matrices 4x1):
The system of equations Ax = c has solutions x = (-4, 4, -3, -2) + p(2, 1, -3, -4) + q(2, -3, 3, -4). The solution set of the system Ax = 0 consists of all vectors x = ?

Question

Absolutely stumped on this one (all matrices 4x1):
The system of equations Ax = c has solutions x = (-4, 4, -3, -2) + p(2, 1, -3, -4) + q(2, -3, 3, -4). The solution set of the system Ax = 0 consists of all vectors x = ?

ganeshie8 · Answer

Hey

anonymous · Answer

@ganeshie8 hi he or she ask me to help lol just playin

clarence · Answer

I'm thinking that it could be x = p(2, 1, -3, -4) but I could be way wrong..

ganeshie8 · Answer

good thinking, remember that the complete solution for $Ax=c$ is given by a particular solution and the complete nullspace :

$$\large {X_{\text{complete}} ~~= ~~X_{\text{particular}} + X_{\text{null}}}$$

ganeshie8 · Answer

In the present problem

$$\large  X_{\text{complete}} = (-4, 4, -3, -2) + \color{red}{p(2, 1, -3, -4) + q(2, -3, 3, -4)}$$

ganeshie8 · Answer

that entire red part constitutes the nullspace
that is, the solutions to the system $Ax=0$

anonymous · Answer

@ganeshie8 lol don't be blowing up my notification k

clarence · Answer

Oh right, that makes sense! I think I might've forgotten about the Xcomplete  =  Xparticular + Xnull when my lecturer was explaining about it, thanks a lot, really appreciate the help!

ganeshie8 · Answer

np :)
here is a little challenge : try finding the matrix $A$ from the given solutions