# Thread: Prob not real tough, but having troubles...

1. ## Prob not real tough, but having troubles...

Hey guys, I'm new to this forum so Hi to everyone. I have a few problems that I"m really having trouble with...I'll start with this one hopefully you guys can help me out.
I want to find an A mxn matrix such that f is linear:
Ax = F(x)

And what I've done:
Suppose:
(a11 a12) * x1 = f1(x)

(a21 a22) * x2 = f2(x)
(This is supposed to be Ax = b) a 2x2 times a 1x2 i just dont kno how to make matrices here)

x1 and x2 can be anything so set x2 = 0
get:
a11*x1 = f1(x1)

a21*x1 = f1(x1)

My professor recommends I now solve for a11, a21 and then the others to construct the A matrix so that f is linear. I have no idea what to do! He says f(x1) can't be anything because it needs to be linear. Please help!
Edit/Delete Message

2. Originally Posted by exostratics
Hey guys, I'm new to this forum so Hi to everyone. I have a few problems that I"m really having trouble with...I'll start with this one hopefully you guys can help me out.
I want to find an A mxn matrix such that f is linear:
Ax = F(x)

And what I've done:
Suppose:
(a11 a12) * x1 = f1(x)

(a21 a22) * x2 = f2(x)
(This is supposed to be Ax = b) a 2x2 times a 1x2 i just dont kno how to make matrices here)

x1 and x2 can be anything so set x2 = 0
get:
a11*x1 = f1(x1)

a21*x1 = f1(x1)

My professor recommends I now solve for a11, a21 and then the others to construct the A matrix so that f is linear. I have no idea what to do! He says f(x1) can't be anything because it needs to be linear. Please help!
Edit/Delete Message
Wouldnt that happen for any m x n matrix?

For any m x n matrix A,isnt the map x $\rightarrow$ Ax (where x is a column vector) a linear map?

3. ## I don't know...

I don't know, but I do know he wants me to specifically solve out for the A matrix.

4. Originally Posted by exostratics
I don't know, but I do know he wants me to specifically solve out for the A matrix.

You should be more clear. If x is a column vector(n x 1), then for any m x n matrix A, F(x) = Ax is a linear map.

Proof:
Vector Addition: F(x1) + F(x2) = Ax1 + Ax2 = A(x1 + x2) = F(x1+x2).
Scalar Multiplication: F(ax) = A(ax) = a(Ax) = aF(x).

Thus F is a linear map.