Let A∈ Μ4,3
(that is, a 4 x 3 matrix). Let v1, v2 ∈ R^4 and let w = v1 +v2. Suppose there exists u1, u2 as an element of R^3 such that v1 = Au1 and v2= Au2
prove the Ax=w is consistent

Question

Let A∈ Μ4,3
(that is, a 4 x 3 matrix). Let v1, v2 ∈ R^4 and let w = v1 +v2. Suppose there exists u1, u2 as an element of R^3 such that v1 = Au1 and v2= Au2
prove the Ax=w is consistent

cruffo · Answer

Question: True of False
A matrix equation, Ax= b, is "consistent" if it has at least one solution.

anonymous · Answer

true

cruffo · Answer

Ok... we seem to be given alot to work with.  

We're told that we can assume that there exists a u1 and u2 so that
v1 = Au1 and v2 = Au2
Sooo...
w = v1 + v2 =  Au1 + Au2

cruffo · Answer

Question:
Au1 + Au2 = A(u1 + u2) ???

anonymous · Answer

that's what i wrote down...

cruffo · Answer

Right.  If    Au1 + Au2 = A(u1 + u2), then just take x = u1 + u2 as a solution.

To convince yourself that  Au1 + Au2 = A(u1 + u2), do an "element-by-element" proof

anonymous · Answer

that's all i had to do?

cruffo · Answer

I think so.  They may be trying to be tricky with the dimensions. A has to act on a vector in R3, and will return a vector in R4.  So the dimensions look fine.

anonymous · Answer

ok  thanks