Problem 15 - Differential Equations and Linear Algebra

Question

Determine \(T(\mathbf{v})\) for the given linear transformation \(T\) and vector in \(V\) by (a) Computing \([T]_{B}^{C}\) and \([\mathbf{v}]_{B}\) and using Theorem 6.5 .4 (b) Direct calculation. \(T: P_{3}(\mathbb{R}) \rightarrow \mathbb{R}\) via \(T(p(x))=p(2),\) relative to the standard bases \(B\) and \(C ; p(x)=2 x-3 x^{2}\).

Step-by-Step Solution

Answer

(a) Using Theorem 6.5.4, we find [T]_B^C = [1, 2, 4, 8] and [v]_B = [0, 2, -3, 0]. Computing T(v) = [T]_B^C[v]_B, we get T(v) = -4. (b) Direct calculation of T(p(x)) = p(2) = 2(2) - 3(2)^2 = -4. Both methods yield T(v) = -4.

Step 1: a) Computing [T]_B^C and [v]_B using Theorem 6.5.4

First, let's find the matrix representation of T. Since T is a linear transformation from P_3 to R, we know its matrix will have dimensions 1x4. We need to find T applied to each basis vector in B. The standard basis for P_3 is given by B = {1, x, x^2, x^3}. The linear transformation T is defined as T(p(x)) = p(2): T(1) = 1(2) = 1 T(x) = 2 T(x^2) = 2^2 = 4 T(x^3) = 2^3 = 8 Thus, [T]_B^C = [1, 2, 4, 8]. Now let's find the coordinates of the vector v in the basis B. We represent p(x) = 2x - 3x^2 in the standard basis for P_3. Therefore, [v]_B = [0, 2, -3, 0]. To find T(v), we use the theorem 6.5.4 which states Tv = [T]_B^C[v]_B: T(v) = [1, 2, 4, 8] [0, 2, -3, 0]^T = 1*0 + 2*2 + 4*(-3) + 8*0 = -4

Step 2: b) Direct calculation

For direct calculation, let's find T(p(x)). By definition of the linear transformation T, we have: T(p(x)) = p(2) = 2(2) - 3(2)^2 = 4 - 12 = -4 Now we have found T(v) using both methods, and they match: T(v) = -4.

Key Concepts

Matrix Representation

The matrix representation of a linear transformation is a key concept in linear algebra. It involves expressing the transformation in terms of a matrix. In this exercise, we're looking at the transformation from the polynomial space of degree 3, denoted as \( P_3(\mathbb{R}) \), to the real numbers \( \mathbb{R} \). This means our task is to find a matrix that effectively "captures" the transformation behavior.

• Here, the transformation \( T \) is defined as \( T(p(x)) = p(2) \).
• The matrix representation, \([T]_B^C\), is derived by applying the transformation to each of the standard basis vectors in \( B \).
• Given \( B = \{1, x, x^2, x^3\} \), applying \( T \) results in the matrix \([1, 2, 4, 8]\).

Understanding the construction of this representation simplifies working with transformations, especially when those transformations need to be calculated across different contexts or vectors.

Standard Basis

The standard basis plays a pivotal role in linear algebra, especially in polynomial spaces. When dealing with polynomials of degree up to 3, like in this exercise, the standard basis is given by \( \{1, x, x^2, x^3\} \).

• These basis vectors are the building blocks or fundamental components of any polynomial in \( P_3(\mathbb{R}) \).
• The transformation \( T \) is applied to each of these basis vectors to form the matrix representation of \( T \).
• Knowing the standard basis allows you to express any polynomial as a combination of the basis vectors, making computations more structured and systematic.

This foundation is crucial in linear transformations as it provides a reference point from which other vectors and transformations can be understood and calculated.

Polynomial Functions

Polynomial functions are expressions consisting of variables raised to powers and multiplied by coefficients. In this exercise, we work with the polynomial \( p(x) = 2x - 3x^2 \).

• Understanding how to evaluate and transform polynomial functions is essential, especially when transformations map them into different spaces or forms like \( \mathbb{R} \).
• The given transformation \( T(p(x)) = p(2) \) evaluates the polynomial at a specific point \( x = 2 \).
• By analyzing how \( T \) affects each term of the polynomial, we confirm that it transforms \( p(x) \) into specific real numbers as seen in the exercise computations.

Polynomials are versatile and appear in numerous mathematical contexts, from calculus to differential equations, making them important to understand deeply.

Linear Algebra Theorems

Theorems in linear algebra provide the foundation for understanding and solving complex problems. They offer shortcuts and frameworks for computations. For instance, the theorem used in our exercise, Theorem 6.5.4, is utilized to compute the result of a transformation.

• The theorem states that the transformation of a vector \( \mathbf{v} \) through \( T \) can be determined using the matrix representation \([T]_B^C\) and the vector's representation \([\mathbf{v}]_B\).
• In the exercise, this theorem simplifies the task because once \([T]_B^C\) and \([\mathbf{v}]_B\) are known, you perform simple matrix multiplication to find \( T(v) \).
• The solution shows that regardless of the calculation method (matrix representation or direct evaluation), the result \( T(v) = -4 \) remains consistent.

These theorems offer a robust framework for understanding complex transformations and ensuring that computations within linear algebra are coherent and reliable.