Need to show that Ax=w is consistent

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** = A**u1** and **v2** = A**u2** for some vectors **u1** and **u2** in R4. What fact allows you to conclude that the system A**x** = **w** is consistent?

What I have calculated so far based on algebraic rules for matrices:

**w** = **v1** + **v2 = **A**u1 + **A**u2 **= 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?

Re: Need to show that Ax=w is consistent

Quote:

Originally Posted by

**Chaobunny** Does anyone have some tips or hints?

Not necessary, your argument is correct: $\displaystyle u_1+u_2$ is a solution of $\displaystyle Ax=w$ so, the system is consistent.

Re: Need to show that Ax=w is consistent

Ah, is that it? That certainly simplifies things-- I guess I was overthinking it. Thank you.

Re: Need to show that Ax=w is consistent

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.