Which of the sets that follow are spanning sets for $R^3?$ Justify your answers.

$($a$)\{(1,0,0{)}^T,(0,1,1{)}^T,(1,0,1{)}^T\}$

$($b$)\{(1,0,0{)}^T,(0,1,1{)}^T,(1,0,1{)}^T,(1,2,3{)}^T\}$

$($c$)\{(2,1,-2{)}^T,(3,2,-2{)}^T,(2,2,0{)}^T\}$

$($d$)\{(2,1,-2{)}^T,(-2,-1,2{)}^T,(4,2,-4{)}^T\}$

$($e$)\{(1,1,3{)}^T,(0,2,1{)}^T\}$