Answer
See below.
Work Step by Step
$T(a+b)=(a+b)''-16(a+b)=a''+b''-16a-16b=T(a)+T(b)$
$T(ca)=(ca)''-16(ca)=c\cdot a''+c\cdot (-16)a=cT(a)$
Thus it is a linear transformation.
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