Question

Determine if there exists a linear transformation T: R2 -> R2 with the following properties.

If yes, give an example. If no, explain why such a transformation is not possible.

(4) Determine if there exists a linear transformation T: R2 + R2 with the following properties. If yes, give an example. If no, explain why such a transformation is not possible. (a) T is one-to-one and onto. (b) T is not one-to-one. (c) T is not onto. (d) T is one to one but not onto. (e) T is onto but not one to one.

Answer 1

I: R²_1R² lineaa transformation, (a) Tis one-one & onto T(x, y) = (0, y le identity transformation is an Iasiret example of one-one & onto L.T. I is not one-one. - Tln, y) = (x,x) then this transformation is not one one because for og T (2, 1) =(2, 2) (2 4 ) = (2, 2) I (1, 2) = (2, 2) so two different elements have some image I I (C) - T is not onto T(2,4) = (x2) then this transformation is not onto because the members (x, y) where x 74 - will not have its preimage in the IR². So Range & co-domain ! NOT INTO

Tis one-one but not onto There is no linear transformation from IR² - 1K² that is one-one but not onto because a L'T from IR IR" is if and only if it is onto one-one Tis onto but not one-one. existe No Linear transformation Reason same eis Cd).