I have some problems, help
Solve the following nonhomogeneous system of linear equations:
Please enter the specific solution first and then the basis in the space of solutions of the corresponding homogeneous system.
Comment of the teacher
Example. If is a specific solution and is a basis of then please enter
There is an automated system here, checking the solution.
I managed to transform the augmented matrix to the diagonal form:
So, the specific solution is the last row + 0 at the bottom: . I checked, and it is really a solution of the original problem.
But, why does the basis in the comment consist of 2 columns? I think, it must be only 1: - the 4-th column of the diagonal matrix and the last element (-1) is an element of negated identity matrix.
When I tried this solution: [1, -3, 3, 0],[0, -1, 1, -1] for the first time, the automated checker said: "incorrect". Then I wrote this question; but before posting it, I tried it again, and it said: "correct". Probably, I made a typo first time
How can I check, that
is really the basis of the space of solutions, like I did, validating the specific solution?