Explanation of solution to given question (subspace)

1. The problem statement, all variables and given/known data







-4x₁ +2x₂ =2

4x₁ -3x₂ -2x₃ =-3

2x₁ -x₂ +(k - k²)x₃ =-k



Find the values of k for which the system has 1) unique solution, 2) infinitely many solution and 3) no solution







3. The attempt at a solution



In REF: the matrix is

-4 2 0 | 2

0 -1 -2 | -1

0 0 k(1-k) | 1-k





1) I do not understand why the solution sets x₃ = y to give a single parameter solution set.





2) for a system to have infinitely solution k must be 1 because 1(1-1) = 0 and 1-1 = 0

0 = 0 implies infinitely many solution



3) for a system to have no solution, the solution must be inconsistent

and so, k(1-k) must equal to 0 for some value k and 1-k gives a value that is an element of real number with the exclusion of 0. For this to be true, k = 0.

0(1-0) = 0 and 1-0 = 1

we see that 0 = 1 but this produces a contradiction.