Calculating the Change of Basis Matrix for Linear Maps on Polynomials is a topic that often trips up students. We’ll be tackling a problem involving linear maps on polynomials, specifically those of degree at most 3. The core challenge involves understanding how the Change of Basis Matrix impacts the matrix representation of a linear transformation when we’re not using the standard basis. This isn’t just about abstract concepts; it has real-world implications in various applications of linear algebra.

Therefore, we’ll dissect a specific example where a scaled basis is used instead of the standard monomial basis. This seemingly small change significantly affects the final Change of Basis Matrix. We’ll walk through the process of computing the correct matrix representation, highlighting the crucial role of the scaling factor and explaining why a common mistake occurs. By the end, you’ll have a solid grasp of how to handle these types of problems and avoid common pitfalls.

---

---

This blog post delves into a problem concerning the calculation of the change of basis matrix for a linear map acting on the vector space of polynomials of degree at most 3. We will explore the intricacies of basis transformations and matrix representations in linear algebra. This problem highlights the importance of understanding how the choice of basis affects the matrix representation of a linear transformation. Let’s begin!

Problem Statement

Let ##V = P_3(\mathbb{R})## be the vector space of polynomials of degree at most 3 over the real numbers. For a fixed ##\delta > 0##, the set ##\{g_0(x) = \delta^{-1}, g_1(x) = \delta^{-1}x, g_2(x) = \delta^{-1}x^2, g_3(x) = \delta^{-1}x^3\}## forms a basis for ##P_3(\mathbb{R})##. Consider the linear map ##L_s: P_3(\mathbb{R}) \to P_3(\mathbb{R})##, for a fixed ##s \ge 0## and ##\phi \in \mathbb{R}##, defined by:

###L_s f(x) = (\phi s – x^2)f'(x) + \frac{1}{3}f'''(x)###

The matrix representation of ##L_s##, denoted by ##(a_{ij})_{i,j=0,1,2,3}##, satisfies ##L_s g_j = \sum_{i=0}^3 a_{ij} g_i## for all ##j = 0, 1, 2, 3##. We need to determine the correct matrix representation of ##L_s## and resolve the discrepancy between two different sources.

Solution

Understanding the Problem

The core of this problem lies in understanding how the choice of basis affects the matrix representation of a linear transformation. We are given a linear map ##L_s## acting on polynomials, and two different matrices are proposed as its representation. The discrepancy arises from the use of a scaled basis ##\{g_i\}## instead of the standard monomial basis ##\{1, x, x^2, x^3\}##. The key to resolving this is to carefully compute the action of ##L_s## on each basis element ##g_i## and express the result as a linear combination of the ##g_i##’s. The coefficients of this linear combination will form the columns of the matrix representation.

The problem involves a subtle point in linear algebra: while the change of basis matrix between the standard monomial basis and the scaled basis is diagonal, the action of the linear map is not invariant under this scaling. This means that the matrix representation will change depending on the basis used. This is a crucial concept in understanding how linear transformations are represented in different coordinate systems. The seemingly simple scaling of the basis vectors introduces a non-trivial change in the matrix representation of the linear map. We need to account for this scaling when computing the matrix representation with respect to the scaled basis.

Solving the Problem

Step 1: Applying the Linear Map to the Basis Elements

Let’s compute the action of ##L_s## on each basis element:

• ##L_s g_0(x) = L_s(\delta^{-1}) = 0##
• ##L_s g_1(x) = L_s(\delta^{-1}x) = (\phi s – x^2)(\delta^{-1}) = \phi s \delta^{-1} – \delta^{-1}x^2 = \phi s g_0 – g_2##
• ##L_s g_2(x) = L_s(\delta^{-1}x^2) = (\phi s – x^2)(2\delta^{-1}x) = 2\phi s \delta^{-1}x – 2\delta^{-1}x^3 = 2\phi s g_1 – 2g_3##
• ##L_s g_3(x) = L_s(\delta^{-1}x^3) = (\phi s – x^2)(3\delta^{-1}x^2) = 3\phi s \delta^{-1}x^2 = 3\phi s g_2##
• ##L_s g_3(x) = L_s(\delta^{-1}x^3) = (\phi s – x^2)(3\delta^{-1}x^2) = 3\phi s \delta^{-1}x^2 = 3\phi s g_2##

Notice that the derivatives of the basis functions are also scaled by the same factor ##\delta^{-1}##. This scaling factor is crucial and is the source of the discrepancy.

Step 2: Constructing the Matrix Representation

From the above calculations, we can construct the matrix representation of ##L_s## with respect to the basis ##\{g_0, g_1, g_2, g_3\}##:

### [L_s] = \begin{pmatrix} 0 & \phi s & 0 & 0 \\ 0 & 0 & 2\phi s & 0 \\ 0 & -1 & 0 & 3\phi s \\ 0 & 0 & -2 & 0 \end{pmatrix} ###

Note that the entries are coefficients of the basis elements in the linear combinations of the results from Step 1. The matrix is expressed in terms of ##\delta^{-1}##. This is the correct matrix representation, resolving the discrepancy.

Step 3: Addressing the Discrepancy

The discrepancy in the source material likely arises from a misunderstanding of how the scaling factor ##\delta## affects the matrix representation. The second proposed matrix includes an additional factor of ##\delta##. This is incorrect. The matrix representation should be consistent with the calculations in Step 2, which correctly accounts for the scaling of the basis vectors. The transformation matrix between the standard monomial basis and the scaled basis is a scalar multiple of the identity, which does commute with other matrices. However, this does not imply that the matrix representation of the linear map is the same in both bases. The linear map itself is affected by the scaling.

Change of Basis Matrix

The change of basis matrix from the standard monomial basis ##\{1, x, x^2, x^3\}## to the scaled basis ##\{g_0, g_1, g_2, g_3\}## is a diagonal matrix with entries ##\delta^{-1}##. Let’s denote this matrix as ##P##. Then, the matrix representation of ##L_s## in the standard monomial basis, denoted by ##[L_s]_M##, is related to the matrix representation in the scaled basis, ##[L_s]_G##, by the equation: ##[L_s]_M = P^{-1}[L_s]_G P##. Since ##P## is diagonal, this simplifies the calculation. However, the key point is that the matrix representation changes when we change the basis, even if the change of basis matrix is simple.

Final Solution

The correct matrix representation of the linear map ##L_s## with respect to the given scaled basis is:

### [L_s] = \begin{pmatrix} 0 & \phi s & 0 & 0 \\ 0 & 0 & 2\phi s & 0 \\ 0 & -1 & 0 & 3\phi s \\ 0 & 0 & -2 & 0 \end{pmatrix} ###

The discrepancy in the other source is due to an incorrect handling of the scaling factor ##\delta## in the basis vectors.

Additional Problems

Problem 1:

Consider the linear transformation ##T: \mathbb{R}^2 \to \mathbb{R}^2## defined by ##T(x, y) = (x + y, x – y)##. Find the matrix representation of ##T## with respect to the standard basis and the basis ##\{(1, 1), (1, -1)\}##.

Solution: The matrix representation with respect to the standard basis is ##\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}##. The matrix representation with respect to the second basis requires a change of basis calculation.

Problem 2:

Let ##T: P_2(\mathbb{R}) \to P_2(\mathbb{R})## be defined by ##T(p(x)) = p'(x)##. Find the matrix representation of ##T## with respect to the basis ##\{1, x, x^2\}##.

Solution: The matrix representation is ##\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 2 \\ 0 & 0 & 0 \end{pmatrix}##.

Problem 3:

Find the change of basis matrix from the basis ##\{1, x, x^2\}## to the basis ##\{1 + x, x, x^2 – 1\}## in ##P_2(\mathbb{R})##.

Solution: Express each new basis vector as a linear combination of the old basis vectors and form the matrix from the coefficients.

Problem 4:

Let ##V## be a vector space with basis ##B = \{v_1, v_2, v_3\}##. Let ##T: V \to V## be a linear transformation such that ##T(v_1) = v_1 + v_2##, ##T(v_2) = v_2 + v_3##, and ##T(v_3) = v_3##. Find the matrix representation of ##T## with respect to ##B##.

Solution: The matrix is formed directly from the coefficients of the transformed basis vectors.

Problem 5:

Verify that if ##P## is the change of basis matrix from basis ##B_1## to basis ##B_2##, then ##P^{-1}## is the change of basis matrix from ##B_2## to ##B_1##.

Solution: This is a fundamental property of change of basis matrices; a direct proof involves matrix multiplication.

---

---