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Question: we call a function t rm rn a...

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We call a function T : Rm → Rn a linear transformation if it satisfies: (1) T(2+ j) T) T( for all vectors , E R; and (2) T(kT) = kT(x) for all vectors x E Rm and all scalars k E R. (Note that this definition differs from the one given in Section 2.1 of the textbook.) Problem 3. (a) Prove that for every function f : R-+ R, if f(cr)-cf(x) for all c E R and x R, then f(a y)f( f(y) for all z, y E R. (In other words, prove that every function f : R- R that preserves scalar multiplication is a linear transformation from R to R.) (b) Give an example to show that the argument you gave in part (a) cannot work in 2 dimen- sions. That is, explicitly describe a function f : R2-ל R2 that is not a linear transformation but has the property that f(cr-cf(a) for all i R2 and c є R. Remember to prove that your example works!

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