vital question!

[Sarahisme]

this is a vitally important question!

 

any hints?

[Dave]

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

[uncool]

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

-Uncool-

[Nicoco]

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.

[Sarahisme]

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

[Sarahisme]

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

[Sarahisme]

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?

[Sarahisme]

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

[Sarahisme]

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

[Nicoco]

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.