If there is a 5 by 5 matrix. rank=2. 2 pivot columns and 3 free columns. To find the null space, we find special solutions 1st, reduce the matrix and then set one free variable to be 1, the other two to be 0s. so Null space = span(three special solutions)... what about the particular solution? Should I set one free variable to be 0, and the other two to be 1s, and then the span of these is particular solution? or just set all these 3 variables to be 0s, and then find the unique one?

Question

If there is a 5 by 5 matrix. rank=2. 2 pivot columns and 3 free columns. To find the  null space, we find special solutions 1st, reduce the matrix and then set one free variable to be 1, the other two to be 0s. so Null space = span(three special solutions)...    what about the particular solution? Should I set one free variable to be 0, and the other two to be 1s, and then the span of these is particular solution? or just set all these 3 variables to be 0s, and then find the unique one?