Please include as the explanation to help me understand each

Please include as the explanation to help me understand each step please.

Let be a linear transformation taking R^n rightarrow R^n. Show that (0) = 0.

Solution

Let T be a linear transformation from R^n to R^n

0 is the additive identity of R^n

We have to prove that 0 is mapped under T to 0 itself

Proof:

We prove this by contradiction

If possible let T(0) not equal to 0

Then since 0 is identity of R^n

0+a = a+0 =a for all a.

T(a+0) = T(0+a) = T(a)

T(a)+T(0) = T(0)+T(a) = T(a)

Hence it follows that T(0) is the identity in the range

Thus we get a contradiction

Hence T(o) = identity = 0 is proved