I'm working through some homework problems right now, and I'm a little confused. Here is the question that I am stuck on.
Determine whether the following maps are linear transformations from $C[0,1]$ to $\mathbb{R}$.
a) $T_{1}(f) = f(0)$
b) $T_{2}(f) = |f(0)|$
c) $T_{3}(f) = \frac{1}{2}(f(0) + f(1))$
d) $T_{4}(f) = \left[\int_{0}^{1}(f(x))^{2} d x\right]^{2}$
It should be noted that $C[0,1]$ is the set of all continuous real-valued functions defined on the interval $[0,1]$, and that $f \in C[0,1]$.
So I know that to prove whether or not something is a linear transformation the following two properties need to hold.
- $T(u + v) = T(u) + T(v), \forall u, v \in V$
- $T(\lambda u) = \lambda (T(u)), \forall \lambda \in \mathbb{F} \text{ and } \forall u \in V$
I understand that each $T$ is just a map of some map in the set $C[0,1]$, but what I don't understand is what exactly $f(0), f(1)$, and $f(x)$ denote in the context of this question? I guess I'm more so looking for clarification as to what exactly is being asked in the above question.
Your maps are maps from $C[0,1]$ to $\mathbb{R}$. Their input is a whole function $f$ in $C[0,1]$ and their output is a real number somehow related to the given function (its value at some point, its integral, etc.) Suppose you want to show property (1) for your first map. You would do $$ T(f+g)=(f+g)(0)=f(0)+g(0)=T(f)+T(g) $$ the first equality being true by definition of $T$ the second by definition of $f+g$ and the third again by definition of $T$