Question

Problem 2. Determine whether the followings transformations are one-to-one, or onto, or both, or neither 418 8r2 (2) T : Rs → R, 지미 ) 1 -22 |12-2x3 r3, r1-2r2

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Answer 1

T: R^2 rightarrow R^2 T([x_1 x_2]) = [x_1 - 2x_2 4x_1 - 8x_2] = [1 4 - 2 - 8] [x_1 x_2] Therefore [1 4 - 2 - 8] almostequalto by R_2 - 4R_1 [1 0 - 2 0] For any choice of vector in R^2 we get here infinite solution Because x_1 - 2x_2 = b Therefore T is not one to one but it is onto [Def^n: one - to - one A transformation T: R^n rightarrow R^m is one to one if for every vector b in R^m the equation T(x) = b has at most one solution x in R^n] Def^n: - ontp: - A transformation T: R^n rightarrow R^m is onto if for every vector b in R^m the equation T(x) = b has atleast one solution x in R^n T: R^3 rightarrow R^2 T([x_1 x_2 x_3] = [x_1 - 2x_2 x_2 - 2x_3] Therefore [1 0 - 2 1 0 - 2] [x_1 x_2 x_3] = [b_1 b_2] Rightarrow [1 0 0 1 - 4 - 2] [x_1 x_2 x_3] = [b_1 - 2b_2 b_2] Therefore for any choice of b_1 and b_2 we get more than one solution Therefore T is not one - one, but it is onto