Question

Let $f\ne 0$ be a fixed polynomial of degree $m\ge 1$. If $p$ is any polynomial, recall that $(p\circ f)(x)=p[f(x)].$ Define $T_f:P_n\to P_{n+m}$ by $T_f(p)=p\circ f$ a. Show that $T_f$ is linear. b. Show that $T_f$ is one-to-one.

Short answer

$T_f$ is linear and one-to-one because it satisfies linearity properties and distinct polynomials $p$ and $q$ cannot have the same $p(f(x))$.

Step-by-step solution

Define Linearity

A transformation is linear if for any polynomials $p(x)$ and $q(x)$, and scalar $c$, the following properties hold:1. $T_f(p+q)=T_f(p)+T_f(q)$2. $T_f(cp)=c{T}_f(p)$.We need to prove these for $T_f$.

Prove Addition Property

Consider $T_f(p+q)$. We have: $$T_f(p+q)=(p+q)\circ f=(p+q)(f(x))=p(f(x))+q(f(x))=T_f(p)+T_f(q).$$Thus, the addition property is satisfied.

Prove Scalar Multiplication Property

Consider $T_f(cp)$. We have:$$T_f(cp)=(cp)\circ f=(cp)(f(x))=c(p(f(x)))=c(T_f(p)).$$Hence, the scalar multiplication property is confirmed, proving $T_f$ is linear.

Define One-to-One Property

A transformation $T_f$ is one-to-one if $T_f(p)=T_f(q)$ implies $p=q$. We need to prove this holds for $T_f$.

Assume Equality and Analyze

Assume $T_f(p)=T_f(q)$. Then:$$p(f(x))=q(f(x)).$$Since $f(x)$ is a fixed polynomial and is not zero, the equality $p(f(x))=q(f(x))$ implies $p(x)=q(x)$ when compared as polynomials, because polynomial identity implies the coefficients must match.

Conclude One-to-One Property

Given $p(f(x))=q(f(x))$ leads to $p(x)=q(x)$, $T_f$ is one-to-one. No two different polynomials $p$ and $q$ can map to the same polynomial under $T_f$.

Key concepts

Polynomial functions

Polynomial functions are expressions that involve variables raised to whole-number exponents and multiplied by coefficients. They can be written in the form: $a_n{x}^n+a_{n-1}{x}^{n-1}+\dots +a_{1}x+a_0$, where each coefficient $a_i$ is a constant, and $x$ is the variable. The function $f(x)$ defines the rule that assigns to each input $x$ an output calculated based on these operations.
When we refer to polynomial functions, we're typically considering functions that map real numbers to real numbers, though they can also take complex values as inputs or outputs. Understanding how polynomials work is crucial since they lay the foundation for more advanced mathematical concepts and have applications in various fields, from physics to economics.
Polynomials can be added, subtracted, multiplied, and composed with each other, creating vast possibilities for calculus and algebra. The power and beauty of polynomials are in both their simplicity and their application across numerous mathematical problems.

Degree of polynomials

The degree of a polynomial is the highest power of the variable in the polynomial expression. It tells you the greatest exponent in the polynomial and gives insight into the function's graph behavior and complexity. For instance, in the polynomial $3x^4+2x^3+x^2$, the degree is 4, because the term $x^4$ is the highest power.
The degree of a polynomial reveals several properties:

• The number of potential roots or solutions the polynomial can have.
• The turning points or vertices of the graph.
• The end behavior of the polynomial function graph as it approaches infinity.

When performing polynomial operations like addition or multiplication, the degree helps determine the degree of the resulting polynomial. It's essential when considering a linear transformation, as the transformation can change but will always add up based on the original degrees in the transformation process.

One-to-one transformations

A one-to-one transformation is a type of bijection, where each element in the domain maps to a unique element in the codomain. For linear transformations like $T_f$ defined by polynomial compositions, proving one-to-one-ness involves showing that if $T_f(p)=T_f(q)$, then the original polynomials $p$ and $q$ must be identical.
More formally, if $T_f(p)$ maps $p$ to a unique polynomial in $P_{n+m}$, it implies there are no two different polynomials that result in the same transformation outcome. This ensures the transformation preserves the distinctiveness of elements in the domain, which can be crucial for solving equations and understanding functions' behaviors.
In polynomials, this is generally achieved because polynomials, when equal, must have equal coefficients for corresponding terms, especially under composition with a non-zero polynomial like $f(x)$. Thus, the nature of polynomial equations supports the enforcement of one-to-one transformations.

Polynomial composition

Polynomial composition involves plugging one polynomial into another, creating a new polynomial where the variable of the outer polynomial is replaced with the entire inner polynomial. If you have $p(x)$ and $f(x)$, the composition $(p\circ f)(x)$ results in the polynomial $p(f(x))$.
Polynomial composition is associative but not commutative, meaning $(p\circ (q\circ r))(x)=((p\circ q)\circ r)(x)$, but typically$(p\circ q)(x)eq(q\circ p)(x)$. This property makes polynomial composition useful for constructing complex functions from simpler building blocks.
For transformation $T_f$, which relies on this concept, understanding composition is pivotal because it allows you to explore dynamic behaviors of polynomials as they transform through mappings. As the degrees of polynomials compose, the resulting polynomial inherits a degree which is the sum of the degrees of the composing functions.

Properties of linear transformations

Linear transformations serve as a central concept in linear algebra, characterized by two main properties:

• Additive Property: For transformations $T$, $T(p+q)=T(p)+T(q)$, showing that the transformation of a sum of polynomials equals the sum of their transformations.
• Scalar Multiplication: $T(cp)=cT(p)$, illustrating that scaling a polynomial and then transforming is the same as transforming first and then scaling.

These properties ensure that linear transformations preserve the operations of vector addition and scalar multiplication. In the context of polynomials, the transformation $T_f$ driven by polynomial composition ($p\circ f$) maintains these properties, confirming its linearity.
Understanding these properties is essential because it allows mathematicians and engineers to simplify complex structures, find solutions efficiently, and predict polynomial behavior or transformations in different contexts. Knowing linear transformations' properties enhances the ability to design problem-solving strategies and mathematical models.