Question: Lete 1 = (1,0,0) T , e 2 = (0,1, 0) T , e 3 = (0,0,1) T be the standard basis of R 3
Lete1= (1,0,0)T, e2= (0,1, 0)T, e3= (0,0,1)Tbe the standard basis ofR3. Consider a linear mapT:R3->R3satisfying T(x, y, z)T= (0,0,0)Twhenever 2x-y+z= 0.
(a) Show thatT ((1/2, 0, -1)T)= (0,0,0)Tand T(0, 1, 1)T= (0, 0, 0)T.
(b) Justify whetherTis an isomorphism using result from (a).
(c) GivenT(e3) = (1,2,3)T. FindT(e1) andT(e2) by using result from (a).
(d) Use your results ofT(e1), T(e2), T(e3) to determine the image of an arbitrary vector
(x, y, z)T, an element ofR3under T.
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