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Automorphism of R3
Task number: 4527
Decide whether the mapping f:R3→R3 given by the formula
f(x,y,z)=(x+y−2z,y−z,x−y)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=(11−201−11−10)
By elementary transformations: (11−201−11−10)∼(11−201−10−22)∼(11−201−1000) we find that its rank is 2 and is therefore singular.
Answer
The mapping f is not an automorphism.