How to determine which of the following transformations are linear transformations? 
Determine which of the following transformations are linear transformations
A. The transformation $T_1$ defined by $T_1(x_1,x_2,x_3)=(x_1,0,x_3)$
B. The transformation $T_2$ defined by $T_2(x_1,x_2)=(2x_1−3x_2,x_1+4,5x_2)$.
C. The transformation $T_3$ defined by $T_3(x_1,x_2,x3)=(x_1,x_2,−x_3)$
D. The transformation $T_4$ defined by $T_4(x_1,x_2,x3)=(1,x_2,x_3)$
E. The transformation $T_5$ defined by $T_5(x_1,x_2)=(4x_1−2x_2,3|x_2|)$.

I believe that it could be A and E. How can I determine this? If someone could show me one I could figure out the rest.

Is T in A a linear transformation?


*

*Check linearity for addition. 


Suppose $T:V \rightarrow W$ .Where $V$ and $W$ are vector spaces over $F$. Let $x_1,x_2,x_3 \in F$ and also let $x_4,x_5,x_6 \in F$. So that $(x_1,x_2,x_3) \in V$ and $(x_4,x_5,x_6) \in V$. Now need to check that $T((x_1,x_2,x_3)) + T((x_4,x_5,x_6)) = T((x_1+x_4, x_2+x_5,x_3+x_6))$. 
We have LHS $ = (x_1,0,x_3) + (x_4,0,x_6) = (x_1+x_4,0,x_3+x_6) = $RHS. Hence this holds by definition of vector addition.


*Check linearity for scalar multiplication:


Let be as above, and suppose $V$ is a vector-space over a field $F$. Then let $a\in F$. Want to prove that:
$aT(x_1,x_2,x_3)=T(ax_1,ax_2,ax_3)$. Hence we have:
LHS $ = a(x_1,0,x_3) = (ax_1,0,ax_3) = $RHS. And this holds by vector scalar multiplication and by property of zero in $\mathbb{R}$.
Hence this is a linear transformation by definition. In general you need to show that these two properties hold.
