Problem 31 - Differential Equations and Linear Algebra

Question

Let \(T: P_{2}(\mathbb{R}) \rightarrow P_{2}(\mathbb{R})\) be the linear transformation satisfying $$T(1)=x+1, \quad T(x)=x^{2}-1, \quad T\left(x^{2}\right)=3 x+2$$. Determine \(T\left(a x^{2}+b x+c\right),\) where \(a, b,\) and \(c\) are arbitrary real numbers.

Step-by-Step Solution

Answer

The short answer to the question is: \(T(ax^2 + bx + c) = b(x^2) + (a + c)(3x) + (a - b + c)(2)\)

Step 1: Apply linear transformation properties

The linear transformation \(T\) has the following properties: 1. \(T(p + q) = T(p) + T(q)\) for all polynomials \(p\) and \(q\) in \(P_2(\mathbb{R})\) 2. \(T(cp) = cT(p)\) for all polynomials \(p\) in \(P_2(\mathbb{R})\) and real numbers \(c\) Using these properties, we can find \(T(ax^2 + bx + c)\) by applying \(T\) to each term in the polynomial and then multiplying by their respective coefficients.

Step 2: Apply T to each term in the polynomial

We need to find \(T(ax^2 + bx + c)\). Using the properties of linear transformations, we can write \(T(ax^2 + bx + c) = aT(x^2) + bT(x) + cT(1)\).

Step 3: Substitute given values for T(1), T(x), and T(x^2)

Now, we can substitute the given values of \(T(1)\), \(T(x)\), and \(T(x^2)\) into the expression. We have: \(T(ax^2 + bx + c) = a(3x + 2) + b(x^2 - 1) + c(x + 1)\)

Step 4: Distribute coefficients and t

Now, distribute the coefficients \(a\), \(b\), and \(c\) to their respective terms: \(T(ax^2 + bx + c) = a(3x) + a(2) + b(x^2) - b(1) + c(x) + c(1)\)

Step 5: Combine like terms

Now, combine all like terms in the expression: \(T(ax^2 + bx + c) = b(x^2) + (a + c)(3x) + (a - b + c)(2)\) Now we have found the polynomial \(T(ax^2 + bx + c)\) for arbitrary real numbers \(a\), \(b\), and \(c\).

Key Concepts

Polynomial Vector Spaces

Polynomial vector spaces are fascinating subjects because they combine the structures of algebra with the infinite dimensionality of function spaces. In particular, we focus on the vector space of polynomials with real coefficients up to a certain degree, denoted as P_n(R). For the vector space of polynomials up to degree 2, which we denote P₂(R), any polynomial can be written in the form a₀ + a₁x + a₂x², where a₀, a₁, and a₂ are real numbers.

One interesting aspect is that these polynomial spaces, much like Rⁿ vector spaces, have bases. For instance, in P₂(R), the set {1, x, x²} forms a basis. This means any second-degree polynomial can be uniquely expressed as a linear combination of these basis polynomials. Moreover, the dimension of P₂(R) is 3, reflecting the three basis polynomials needed to span the space.

Properties of Linear Transformations

Linear transformations play a central role in understanding vector spaces because they preserve the operations of vector addition and scalar multiplication. That is, if T is a linear transformation and u and v are vectors, then T(u + v) = T(u) + T(v). Similarly, for a scalar c, the transformation satisfies T(cu) = cT(u). These properties ensure linearity and are crucial for predicting the behavior of T without having to compute its effects on every element of the vector space.

When applying these properties to polynomial vector spaces, we can simplify computations significantly. If we know the outcome of a linear transformation on a set of basis polynomials, as in our exercise, we can determine its effect on any polynomial in that space by breaking it down into a linear combination of the basis elements. This highlights the interplay between linear transformations and the structure of the vector space they act upon.

Applying Linear Transformations

Applying a linear transformation to polynomials is methodical and utilizes the properties of linearity. Following the steps outlined in the solution, we see a practical application of these rules. The exercise’s transformation T acts on polynomials in P₂(R) and mapping them to other polynomials in the same space.

For T(ax² + bx + c), we applied the transformation to each term individually, adjusted by its coefficient. This approach aligns with the defining properties of linear transformations. By analogously applying the transformation to each basis element and combining the results, we can obtain the output for any polynomial in P₂(R). It's methodical: transforming basis elements, scaling them by respective coefficients, and combining the results, which showcases the inherent systematized nature of linear transformations.