Non homogeneous system question

[nhrock3]

there is a system Ax=b (x and b are vectors)
A is a system of kxn type
the system has at leas one solution:
1.
if k>n does the system has endless solutions
??
2.
if k=n and Ax=b has a single solution then for any system Ax=c has a solution
??


for 1:
i don't know why we need the other data k<n means we have free variables so its endless solutions

for 2:
i don't know what's c

 

[Mark44]

For 1, you are answering a different question than the one given.

For 2, if you are given that k = n and Ax = b has a unique solution, how would you find this solution? How would you find the solution to Ax = c?

 

[bennyska]

one of the ways i think about part 2 is that if Ax=b has a unique solution, then if we take the matrix A and augment it with b, then A can row reduce down to a nxn identity matrix. the augmented part depends on b, certainly, but if A reduces down, then b doesn't really matter, at least considering the row operations required to get the identity. So if we keep track of the row operations we performed on A to get the solution for b, we can perform those same row operations on just any vector c to get the appropriate solution.