##### Practice Midterm SOLUTION: #4
abbyto1607
Since 4 is long and has four parts, I'll only post my answers and you guys can let me know if you think I might've gotten something wrong. If need be, I'll post a more thorough,specific solution. In general, to prove that the transformation is linear, we need to prove that the transformation is closed under both addition and scalar multiplication. In terms of proving, this amounts to testing whether $T((a_{1},b_{1},c_{1}))+T((a_{2},b_{2},c_{2}))=T((a_{1}+a_{2},b_{1}+b_{2},c_{1}+c_{2}))$ and that $kT((a_{1},b_{1},c_{1}))=T((ka_{1},kb_{1},kc_{1}))$ a)linear b)not linear, with the "1" in the last component, T isn't closed under addition or scalar multiplication c)not linear, with the"ab" in the first component, T isn't closed under addition or scalar multiplication d)linear