How do you find a linear transformation with range...?

Question

How do you find a linear transformation with range {( x, y, z, w) Є R^4 : x - y + z - w = 0)} ????

its driving me nuts, thx 4 ne help

Answer 1

Perhaps because the question makes no sense?

What does "x - y + z - w = 0" mean?

If it means that you are looking for a transformation T such that for all 4-tuples ( x, y, z, w), if ( x', y', z', w') = T( x, y, z, w) then
x' - y' + z' - w' = 0, then the answer is easy, because there are many answers - an infinite number of them!

(1,0,0,0)
(0,1,0,0)
(0,0,1,0)
(0,0,0,1)

are a set of basis vectors of R^4. For each of them, choose a T result that satisfies the requirement. It doesn't matter which of the many possibilities you choose:

(0, 0, 0, 0)
(1, 1, 1, 1)
(1, 2, 3, 2)
(1, 0, 0, 1)
etc.

Then, since T is linear, we can define it as:

T( x, y, z, w) = xT(1,0,0,0) + yT(0,1,0,0) + zT(0,0,1,0) + wT(0,0,0,1)

But whether that is really what you are looking for, I can't say. Perhaps you should post the question again once you understand the constraints?

Answer 2

A= -1 1 2 3 2 2 -3 0 -4 4 4 4 if X=(x1,x2,x3,x4) notice that A*x=T(x1,x2,x3,x4) kerT=kerA= {-2 , -1, -2, 1 } rangeT=rangeA=R3