1. ## linear transformation

How do I show that the transformation, T, defined by

Tf(x) = {f(x)d(x) where { is the integral from 0 to 1

is a linear transformation from C, the space of all real-valued continuous functions on [0, 1], into R?

This idea is intuitive, but I'm having a tough time proving it. Thanks for your help.

2. Originally Posted by ktcyper03
How do I show that the transformation, T, defined by

Tf(x) = {f(x)d(x) where { is the integral from 0 to 1

is a linear transformation from C, the space of all real-valued continuous functions on [0, 1], into R?

This idea is intuitive, but I'm having a tough time proving it. Thanks for your help.
Is $\displaystyle d(x)$ fixed? Also, is this $\displaystyle T:\mathcal{C}[0,1]\to\mathbb{R}:f(x)\mapsto\int_0^1f(x)d(x)dx$? What's the problem? For example
$\displaystyle T(f(x)+g(x))=\int_0^1 d(x)(f(x)+g(x))dx=\int_0^1d(x)f(x)+\int_0^1d(x)g(x )$$\displaystyle =T(f(x))+T(g(x))$

3. I think you mean simply $\displaystyle T(f)= \int_0^1 f(x) dx$.

Just use the definition of "linear transformation"- If a and b are real numbers and f and g are in C[0,1] then $\displaystyle T(af+ bg)= \int_0^1 (af(x)+ bg(x)dx= a\int_0^1 f(x)dx+ b\int_0^1 g(x)dx= aT(f)+ bT(g)$.