Questions: Let C0,2 be the space of continuous functions mapping the interval 0 ≤ x ≤ 2 into R. Let T: C0,2 → P be defined by T(f)=∫0^2 f(x) dx. If possible, give three different functions in ker(T).

Let C0,2 be the space of continuous functions mapping the interval 0 ≤ x ≤ 2 into R. Let T: C0,2 → P be defined by T(f)=∫0^2 f(x) dx. If possible, give three different functions in ker(T).

Transcript text: Let $C_{0,2}$ be the space of continuous functions mapping the interval $0 \leq x \leq 2$ into $\mathbb{R}$. Let $T: C_{0,2} \rightarrow \mathbb{P}$ be defined by $T(f)=\int_{0}^{2} f(x) d x$. If possible, give three different functions in $\operatorname{ker}(T)$.

Solution

Solution Steps

To find functions in the kernel of the linear transformation $ T $, we need to identify functions $ f(x) $ such that the integral of $ f(x) $ from 0 to 2 is zero. This means we are looking for functions whose area under the curve from 0 to 2 is zero. Examples of such functions include those that are symmetric about the x-axis over the interval, such as sine or cosine functions with appropriate periods, or piecewise functions that have equal positive and negative areas.

Step 1: Identify Functions in the Kernel

To find functions in the kernel of the transformation $ T $, we need functions $ f(x) $ such that

\[ T(f) = \int_{0}^{2} f(x) \, dx = 0. \]

Step 2: Define the Functions

We consider three functions that satisfy this condition:

$ f_1(x) = \sin(\pi x) $
$ f_2(x) = x - 1 $
$ f_3(x) = (x - 1)(x - 2) $

Step 3: Calculate the Integrals

Now we compute the integrals of these functions over the interval $[0, 2]$:

For $ f_1(x) $: \[ \int_{0}^{2} \sin(\pi x) \, dx = 0. \]
For $ f_2(x) $: \[ \int_{0}^{2} (x - 1) \, dx = 0. \]
For $ f_3(x) $: \[ \int_{0}^{2} (x - 1)(x - 2) \, dx = 0. \]

Final Answer

All three functions $ f_1(x) $, $ f_2(x) $, and $ f_3(x) $ are in the kernel of $ T $ since their integrals over the interval $[0, 2]$ equal zero. Thus, we can conclude:

\[ \boxed{f_1(x) = \sin(\pi x), \, f_2(x) = x - 1, \, f_3(x) = (x - 1)(x - 2)} \]

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