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  3. we call a function rm rn a linear...

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

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We call a function : Rm → Rn a linear transformation if it satisfies: (1) T(F+リ-T(F) +T(j) for all vectors x,y E R; and (2) T(kï) kT() for all vectors a 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(a)-cf(x) for all c R and x є R, then f(x + y)-f(x) + f/(y) for all z,y R. (In other words, prove that every function f : RR 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 R22 that is not a linear transformation but has the property that f(ct) cf() for all ER2 and c E R. Remember to prove that your example works!

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