Answer
See answer below
Work Step by Step
Take $f,g \in C^0[a,b]$ and $c \in R$
we have
$$T(f+g)=\int^b_a (f+g)xdx \\
=\int ^b_a (f(x)+g(x))dx\\
=\int^b_a f(x)dx+\int^b_a g(x)dx\\
=T(f)+T(g)$$
$$T(cf)=\int^b_a (cf)xdx\\
=c\int^b_a f(x)dx\\
=cT(f)$$
Thus it is a linear transformation.
