Question

if u is a solution to Ax = b and v is a solution for Ax = 0 show u + v is a solution to Ax = b

Answer 1

let say \[u=\left[\begin{matrix}u_1\\u_2\\..\\..\\..\\u_n\end{matrix}\right]\] is solution of Ax =b , then \(a_{11}u_1+a_{21}u_2+......+a_{mn}un=b_1\) you still have many lines, but I just pick 1 to show.
and v is the same \(a_{11}v_1+a_{21}v_2+....+a_{mn}v_n =0\)
now add them up, and factor \(a_{ij}\) out
\(a_{11}(u_1+v_1)+a_{21}(u_2+v_2)+.....a_{mn}(u_n+v_n) =b_1\)
do the same with \(a_{12},a_{13}.....a_{1n}\)you will have (u+v) is the solution of Ax =b
I am not sure whether you get my shortcut or not, ok, let me rearrange

Answer 2

I may get ban from this reply. Whatsoever, I don't care, proving problem is not a problem to teach.

Answer 3

thanks alot.