Linear transformation has unique standard matrx? [T]=-I

trefoniu1 2022-01-23 Answered
Linear transformation has unique standard matrx?
[T]=I
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Appohhl
Answered 2022-01-24 Author has 11 answers
Step 1
The statement is false. Any linear transformation satisfies
T(cv)=cT(v)
for all scalars c, as a consequence of linearity. In particular, this is true for c=1, so T(v)=T(v) for all v in Rn
Since all linear transformations satisfy this equation, and there is more than one transformation from Rn to Rn (unless n=0 or R is a trivial ring), therefore it is not true that there is only one linear transformation satsifying this identity. And not a unique standard matrix neither.
The map with standard matrix [T]=I satisfies T(v)=v. This is sometimes called the antipodal map. That defining equation looks a little like the given equation T(v)=T(v), which may explain the confusion. There is indeed a unique linear transformation given by that map (a somewhat tautological statement). So if they had written that equation, their answer would have been correct. But the equation as written, as I said, many linear maps satisfy.
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euromillionsna
Answered 2022-01-25 Author has 16 answers
Step 1
The matrix rel the standard basis for Rn is unique, given a linear transformation T.
Then i- th column is given by T(ei), expressed in terms of the standard basis, where ei is the i- th standard basis vector.
So, there seems to be an error. Indeed, there is only one T=I.
Of course, every linear transformation T satisfies T(v)=T(v), however.
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