Let A be a mn matrix such that Axb and Axc each have a solut

Let A be a m×n matrix such that Ax=b and Ax=c each have a solution for some choice of b and c in R^m. If d=b+c, is the equation Ax=d also consistent?

Hint: Let u and v be solutions to Ax=b and Ax=crespectively,so Au=b and Av=c.Try to combine u and v in some way to get a solution for Ax=d.

Solution

Let u and v be solutions to Ax=b and Ax=crespectively,so Au=b and Av=c

So, Au = b ; Av = c

Adding the two : Au +Av = b+c

Now we can write A( u+v) =b+c

So, u+v is a solution .

A(u+v) = d

Hence Ax = d is also consistent