this is a vitally important question!

 

any hints?

Proof by contradiction seems to be the way to go with this question (indeed, with pretty much every "show this is unique" type question).

Just multiply both sides by C on the left side. Only one x can be the solution to Ix = Cb.

-Uncool-

it's fairly simple i think. Suppose there are two solutions to the system, x and x'. Then both Ax=b and Ax'=b hold. So Ax=Ax'. Multiply by C to get CAx=CAx', which can be reduced to x=x'. So there is only one solution.

thanks guys yeah proof by contradiction is what i tried at first

in understand Nicoco way of doing it, but 'uncool' i don't quite get what you mean?

or is because if Ix = Cb then since the Columns of A are linearly independent, then Ax = 0 has only the trivial solution and therefore it is one-to-one, therefore it is a unique solution?

also if anyone i willing, just quickly, is this right? (for the colmn basis part)

" columnmatr.jpg " is what i think the basis for the column space of matrix A is...

I found this definition of column space: The vector space generated by the columns of a matrix viewed as vectors.

Now looking at your solution, how can a vector with 4 components be a base for 4 vectors with only three components? So if this definition is the right one, I would guess that all the vectors except the zero-vector are the base... I'm going to look this up later to be sure.