Answer to Question #214202 in Linear Algebra for Nanthan

Question

T: R--->R defined by T(x) = x+1, for all x ε R. Is T is

linear?

Answer 1

Solution:

Given, T: R--->R defined by T(x) = x+1, for all x ε R

We need to check following:

(1) T(x + y) = T(x) + T(y) for all x, y ∈ R

(2) T(cx) = cT(x) for all x ∈ R and c ∈ R

For (1): LHS = T(x+y) = (x+y)+1=x+y+1

RHS = T(x)+T(y) = (x+1)+(y+1)=x+y+2

Since, LHS$\ne$ RHS, first property does not hold.

For (2): LHS = T(cx) = cx+1

RHS = cT(x) = c(x+1)= cx+c

Since, LHS$\ne$ RHS, second property does not hold.

Thus, it is not linear.