Question

Let \( T : P_2(\mathbb{R}) \to P_2(\mathbb{R}) \) be the linear transformation defined by \[ T(p(x)) = p(x + 1), \quad \text{for all } p(x) \in P_2(\mathbb{R}) \] If \( M \) is the matrix representation of \( T \) with respect to the ordered basis \( \{1, x, x^2\} \) of \( P_2(\mathbb{R}) \), then which one of the following is TRUE?

• A. The determinant of \( M \) is 2
• B. The rank of \( M \) is 2
• C. 1 is the only eigenvalue of \( M \)
• D. The nullity of \( M \) is 2

Hint:

To find the eigenvalues of a matrix, solve the characteristic equation \( \text{det}(M - \lambda I) = 0 \). For diagonalizable matrices, the eigenvalues are the diagonal elements.

Answer 1

The Correct Option is C

Step 1: Understand the linear transformation.
The linear transformation \( T(p(x)) = p(x+1) \) shifts the polynomial \( p(x) \) by 1. Applying \( T \) to the basis elements \( 1 \), \( x \), and \( x^2 \) gives: \[ T(1) = 1, \quad T(x) = x + 1, \quad T(x^2) = (x+1)^2 = x^2 + 2x + 1 \] Step 2: Find the matrix representation of \( T \).
The matrix representation of \( T \) with respect to the basis \( \{1, x, x^2\} \) is found by writing the images of the basis elements as linear combinations of the basis: \[ T(1) = 1 \cdot 1 + 0 \cdot x + 0 \cdot x^2 \] \[ T(x) = 1 \cdot 1 + 1 \cdot x + 0 \cdot x^2 \] \[ T(x^2) = 1 \cdot 1 + 2 \cdot x + 1 \cdot x^2 \] Thus, the matrix representation of \( T \) is: \[ M = \begin{pmatrix} 1 & 1 & 1
0 & 1 & 2
0 & 0 & 1 \end{pmatrix} \] Step 3: Determine the eigenvalues of \( M \).
The eigenvalues of \( M \) are the roots of the characteristic equation, given by: \[ \text{det}(M - \lambda I) = 0 \] For \( M \), the characteristic equation is: \[ \text{det} \begin{pmatrix} 1-\lambda & 1 & 1
0 & 1-\lambda & 2
0 & 0 & 1-\lambda \end{pmatrix} = (1 - \lambda)^3 = 0 \] This implies that \( \lambda = 1 \) is the only eigenvalue of \( M \), with multiplicity 3. Final Answer: \[ \boxed{1 \text{ is the only eigenvalue of } M} \]