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Chapter 2: Problem 15

Recall the definition of \(\mathrm{P}(R)\) on page 11. Define $$ \mathrm{T}: \mathrm{P}(R) \rightarrow \mathrm{P}(R) \quad \text { by } \quad \mathrm{T}(f(x))=\int_{0}^{x} f(t) d t . $$ Prove that \(\mathrm{T}\) linear and one-to-one, but not onto.

Short Answer

Expert verified
In conclusion, to prove that T is linear, one-to-one, and not onto: 1. Linearity of T is shown by proving \(T(af_1(x) + bf_2(x)) = aT(f_1(x)) + bT(f_2(x))\), which follows from the linearity of integration. 2. One-to-one is proven by showing \(T(f(x))=0\) implies \(f(x)=0\), which is a result of the fundamental theorem of calculus. 3. T is not onto by providing a counterexample, such as \(g(x)=1\), and showing there is no polynomial \(f(x) \in P(R)\) satisfying \(T(f(x))=g(x)\), which results in a contradiction.

Step by step solution

01

Prove the linearity of T

To prove the linearity, we need to show that \(T(af_1(x) + bf_2(x)) = aT(f_1(x)) + bT(f_2(x))\). Let \(f_1(x)\) and \(f_2(x)\) be any polynomials in P(R) and let a and b be any constants in R. Then: \(T(af_1(x) + bf_2(x)) = \int_{0}^{x}(af_1(t) + bf_2(t))dt\) Now, by using the linearity of integration, we can write: \(= a\int_{0}^{x}f_1(t)dt + b\int_{0}^{x}f_2(t)dt\) \(= aT(f_1(x)) + bT(f_2(x))\) which proves that T is linear.
02

Prove T is one-to-one

To prove T is one-to-one, we need to show that if \(T(f)=(0)\), then \(f(x)=(0)\). Consider any polynomial \(f(x) \in P(R)\) such that \(T(f(x))=0\). This means \(\int_{0}^{x} f(t) dt = 0\) Now, let's differentiate both sides with respect to x: \(\frac{d}{dx} \int_{0}^{x} f(t) dt = \frac{d}{dx} 0\) Applying the fundamental theorem of calculus, we get: \(f(x) = 0\) It follows that the only polynomial in the null space of T is the zero polynomial, so T is one-to-one.
03

Prove T is not onto

To prove T is not onto, we need to provide an example of a polynomial g(x) in P(R) that has no pre-image under T, that is, there does not exist any polynomial \(f(x) \in P(R)\) such that \(T(f(x)) = g(x)\). We need to find g(x) such that there is no solution for: \(\int_{0}^{x} f(t) dt = g(x)\) Consider the constant polynomial \(g(x) = 1\). Suppose there is a polynomial f(x) such that: \(\int_{0}^{x} f(t) dt = 1\) Now, differentiate both sides with respect to x: \(\frac{d}{dx} \int_{0}^{x} f(t) dt = \frac{d}{dx} 1\) Applying the fundamental theorem of calculus, we have: \(f(x) = 0\) However, this is a contradiction because the integral of the zero polynomial cannot be equal to 1. Consequently, there is no polynomial f(x) satisfying this equation, and T is not onto. In conclusion, the transformation T is linear and one-to-one, but not onto.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linearity of Integration
To understand the linearity of integration, imagine you are combining ingredients to bake cookies. You might mix sugar and flour separately, but the total amount is the combination of the two. Similarly, integration鈥攅ssentially a mathematical way of combining functions鈥攆ollows the same principle. If we have two functions, say f1(x) and f2(x), and we multiply them by constants a and b, respectively, integrating the sum of these two functions is the same as integrating them separately and then adding the results.

Fundamental Theorem of Calculus
The fundamental theorem of calculus is a game-changer. It's what connects the concept of differentiation, which gives the rate of change, with integration, which deals with the accumulation of quantities. For a continuous function f(x), this theorem allows us to evaluate the integral of f by taking an antiderivative and evaluating it at the endpoints. Simply put, it assures us that if we accumulate small changes (as performed by integration), we can reverse the process and find the rate of change (by differentiation) at any point. This theorem is a key player when proving whether a transformation like T is one-to-one.

One-to-One Transformation
A one-to-one transformation, in the mathematics world, is like a strict guest list at a private event; every guest (output) has a unique invitation (input), and no two guests share the same one. When we say a transformation is one-to-one, we're declaring that for every different input there's a different output. No duplicates allowed! In terms of our function T, if you start with a polynomial and apply T, you should only get zero if you started with the zero polynomial.

Onto Transformation
An onto transformation is like having enough seats for every guest at a concert; even if the guest list is specific, each guest will find a place. For transformations, being onto means that for every polynomial in the target space, there's a polynomial in our starting space that gets transformed into it. If T were onto, every polynomial in P(R) could be traced back to another polynomial that, once put through T, equals it. Through the example in the solution with the constant polynomial g(x) = 1, we saw that no such source polynomial exists for this case, indicating that T isn't onto.

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Most popular questions from this chapter

Let \(\mathrm{T}: \mathrm{R}^{3} \rightarrow R\) be linear. Show that there exist scalars \(a, b\), and \(c\) such that \(\mathrm{T}(x, y, z)=a x+b y+c z\) for all \((x, y, z) \in \mathrm{R}^{3}\). Can you generalize this result for \(\mathrm{T}: \mathrm{F}^{n} \rightarrow F ?\) State and prove an analogous result for T: \(\mathrm{F}^{n} \rightarrow \mathrm{F}^{m}\).

Let $$ \begin{aligned} \alpha &=\left\\{\left(\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right),\left(\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right),\left(\begin{array}{ll} 0 & 0 \\ 1 & 0 \end{array}\right),\left(\begin{array}{ll} 0 & 0 \\ 0 & 1 \end{array}\right)\right\\}, \\ \beta &=\left\\{1, x, x^{2}\right\\}, \end{aligned} $$ and $$ \gamma=\\{1\\} . $$ (a) Define \(\mathrm{T}: \mathrm{M}_{2 \times 2}(F) \rightarrow \mathrm{M}_{2 \times 2}(F)\) by \(\mathrm{T}(A)=A^{t}\). Compute \([\mathrm{T}]_{\alpha}\). (b) Define $$ \mathrm{T}: \mathrm{P}_{2}(R) \rightarrow \mathrm{M}_{2 \times 2}(R) \quad \text { by } \quad \mathrm{T}(f(x))=\left(\begin{array}{cc} f^{\prime}(0) & 2 f(1) \\ 0 & f^{\prime \prime}(3) \end{array}\right), $$ where ' denotes differentiation. Compute \([\mathrm{T}]_{\beta}^{\alpha}\). (c) Define \(\mathrm{T}: \mathrm{M}_{2 \times 2}(F) \rightarrow F\) by \(\mathrm{T}(A)=\operatorname{tr}(A)\). Compute \([\mathrm{T}]_{\alpha}^{\gamma}\). (d) Define \(\mathrm{T}: \mathrm{P}_{2}(R) \rightarrow R\) by \(\mathrm{T}(f(x))=f(2)\). Compute \([\mathrm{T}]_{\beta}^{\gamma}\). (e) If $$ A=\left(\begin{array}{rr} 1 & -2 \\ 0 & 4 \end{array}\right), $$ compute \([A]_{\alpha}\). (f) If \(f(x)=3-6 x+x^{2}\), compute \([f(x)]_{\beta} .\) (g) For \(a \in F\), compute \([a]_{\gamma}\).

$$ \mathrm{T}: \mathrm{R}^{3} \rightarrow \mathrm{R}^{2} \text { defined by } \mathrm{T}\left(a_{1}, a_{2}, a_{3}\right)=\left(a_{1}-a_{2}, 2 a_{3}\right) $$

Let \(V\) and \(W\) be vector spaces, and let \(T\) and \(U\) be nonzero linear transformations from \(\mathrm{V}\) into \(\mathrm{W}\). If \(\mathrm{R}(\mathrm{T}) \cap \mathrm{R}(\mathrm{U})=\\{0\\}\), prove that \(\\{T, U\\}\) is a linearly independent subset of \(\mathcal{L}(V, W)\).

(a) Show that \(D: C^{\infty} \rightarrow C^{\infty}\) is a linear operator. (b) Show that any differential operator is a linear operator on \(C^{\infty}\).
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