Question

Suppose T:R3-R3 is the transformation given below. Determine whether I is one-to-one and/or onto. If it is not one-to-one, show this by providing two vectors that have the same image under T. If T is not onto, show this by providing a vector in R3 that is not in the range of T. хо 3x0-3x4–3x2 3x0-3x1 x2 X0-X1-3x2 TX1 = T is not one-to-one: 0 0 0 0 TO = and TO 0 0 0 0 T is not onto: T is onto T is not onto: IS TIL Le image of any x under T. 0 U

Answer 1

T: R3IR² is giress by T ng 30-394-3% 3xo-324 neorry-axz Let [2] Си-ex T Then T [6] - L 08. 3a3bber 3a 3b a-b-3c ca 91 39-30-3620 Bar 3 beo afb-3020 From 2 9-620 or asb. From o OzCoo od ezo, Tuor eter [1] - [8] -a [3] --- ker T Sp spool Then dime kest) al and kesto] ie T is not one one. We know that every lineas trans- formation image of o be T(0)=0

Therefore two elements of [8] and [6] have same image under T. By Rank- Nurity theorem Rank (T) + Nucity (t) = 3. or, RCT) = 3r Nunity CT) 0 3-1 22. Therefore Tis not onto. How Ruft) - SP Rufo) = spf:[A1 T[!] [81] se{ [] TL 13) Sp(131-13) Since [!] € kest so 1] ¢ Range)