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Automorphism of R3

Task number: 4527

Decide whether the mapping f:R3R3 given by the formula f(x,y,z)=(x+y2z,yz,xy)T is an isomorphism of R3 onto itself (so-called automorphism).
  • Solution

    The matrix of the mapping with respect to the standard basis E of the space R3 (on both sides) is: fE,E=(112011110)

    By elementary transformations: (112011110)(112011022)(112011000) we find that its rank is 2 and is therefore singular.

  • Answer

    The mapping f is not an automorphism.
Difficulty level: Easy task (using definitions and simple reasoning)
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