Linear transformations

[Claire8128]

I’ve tried solving this problem using cT(u) = T(c(u)) to see if the transformations are linear. I thought that the answer was ACE since adding makes them non-linear transformations but I’m not sure how else to solve the problem.

 

[Romsek]

one way of determining whether a transformation is linear is to see if you can come up with a matrix that performs the transformation.

also remember any linear transformation applied to 0 returns 0.

So A is not linear because of the 3 in y2=3.

B is not linear because of the 2 in y1=x1+2

C is indeed linear as are D and E

F is not because of the x1x2 in y2=x1x2

 

[Dr.Peterson]

I’ve tried solving this problem using cT(u) = T(c(u)) to see if the transformations are linear. I thought that the answer was ACE since adding makes them non-linear transformations but I’m not sure how else to solve the problem.

Please show what you did to "use [MATH]cT(u) = T(c(u))[/MATH]".

What I would mean by that is to replace the vector u with cu, and see what happens.

For A, for example, if [MATH]u=[x_1][/MATH] then [MATH]cu = [cx_1][/MATH]. Then the vector [MATH]T(cu)[/MATH] is [MATH][y_1,y_2] = [6(cx_1), 3][/MATH]. What is [MATH]cT(u)[/MATH]? Are they the same?

 

[Steven G]

You can eliminate some choices by using the fact that T(0)=0.

Choice a) y2(0) = 3.. Not linear
Choice b) y1(0) =2 which is not 0.
etc