Thread: infinit number of solutions (lenear akgebra)

question:
there are two systems of MXN non homogeneus equations:
they both are from Ax=b type
and their redused A is the same.
the first system c' has infinit number of solution
does the other one(c) has infinit number of solutions too .
?
i know that its wrong but in this course i have a list of laws regarding systems of equations
and if the answer was "yes" then i should have prove it.
but i dont know why by the laws its impossible???(so i need to look for contradicting example)

As you say, the left hand sides of the two systems are equivalent, because the matrix A in both cases has the same reduced form. So the two systems can only differ on the right hand side, which means that the b's can be different.

The first system might be

Ax = b_1,

and the second system might be

Ax = b_2.

[Edit: The following part has been pointed out to be wrong] Since the first system has an infinite number of solutions, the reduced form of A must contain a row consisting only of zeroes. For the system not to be inconsistent, the corresponding entry in b_1 must also be 0.

If however the corresponding entry in b_2 is non-zero, then the system Ax = b_2 will have no solution.

In conclusion, the two systems need not have the same number of solutions. The number of solutions depend on the vector b.

why the reduced form of A must contain a row consisting only of zeroes ??
there can bu a systems of 5 equations and 3 variables
the reduced form of A will not have a row of zeros
you will have free variables which will give you infinite number of solutions

for example
1 0 0 | 4
1 1 6 |6

Ungh! You are right, of course.

However, in the case that the reduced form of A does contain a row of zeroes, you can use what I said as a counterexample.

yes but i wasnt told that so ...

Weren't you asked, if the other system would also always have an infinite number of solutions, given that the first one had an infinite number of solutions?

The answer to this question is no, as the example with the zero row shows.

yes you are right
thanks

ok another question

the same thing but if the first one has a single solution
then the other one too

??

What if the first system looked like this:

1 0 | 5
0 1 | 7
0 0 | 0

This system has a single solution. But the other system could look like this:

1 0 | 5
0 1 | 7
0 0 | 9,

and this system has no solutions.