Here is the problem I have. It seems like I just have to use some of the properties from the book to solve this but I can't wrap my head around it.
Let A be a 3 x 4 matrix, let v1 and v2 be vectors in R3, and let w = v1 + v2. Suppose v1 = Au1 and v2 = Au2 for some vectors u1 and u2 in R4. What fact allows you to conclude that the system Ax = w is consistent?
What I have calculated so far based on algebraic rules for matrices:
w = v1 + v2 = Au1 + Au2 = A(u1 + u2), so u1 + u2 = x.
I know a system is consistent only if it has a solution, so that's what I'm trying to prove. But this is very abstract so I'm not sure where to begin. I feel like it has something to do with multiplying A by x but I'm not really sure. Does anyone have some tips or hints?
You may have been thinking that you needed to show that the equation had a unique solution. That is not true. A system of equations, or matrix equation, is "consistent" as long as it has a solution. It does not have to be unique. In particular, the equation Ax= 0 always has the trivial solution, x= 0, so is "consistent" no matter what A is.