TITLE: Intuition behind the definition of linear transformation
QUESTION [19 upvotes]: I have studied that given vector spaces $V_1$ and $V_2$, a function $T:V_1 \rightarrow V_2$ is called a linear transformation of $V_1$ into $V_2$, if following two properties are true for all $u, v \in V_1$ and scalar $c$:
$(1)$:  $T(u+v) = T(u) + T(v)$ and $(2):  T(cu) = c T(u) $.
My questions are 
$1$: What is the geometrical interpretation of properties $1$ and $2$ which says that $T$ preserves additivity and scalar multiplication. I am not able to see this geometrically. What is the meaning of preserving additivity and scalar multiplication.
$2$: At some place I have studied that a linear transformation will be linear if it sends each line to line and planes to planes and so on. How can we interpret this based on these two properties.
I need help to understand this. 
Thank you very much for your time..

REPLY [2 votes]: I am giving you the intuitive meaning of linear transformation based oon coordinate geometry because it is easy to understand and we learned it in higher secondary school.
linear transformation A transformation in which the origin of reference frame doesn't change and new cordinate is obtained is linear function of old coordinate i.e. X'=aX+bY and Y'=cX+bY is called linear transformation. In such stated transformation, the straight line remains straight.
You can now replace term old coordinate with standard basis or standard basis vector and new cordinate with cordinate basis or particular vector.