Question

Let P₇(x) be the real vector space of polynomials, in the variable x with real coefficients and having degree at most 7, together with the zero polynomial.Let T : P₇(x) → P₇(𝑥) be the linear transformation defined by
\(T(f(x))=f(x)+\frac{df(x)}{dx}\).
Then, which one of the following is TRUE ?

• A. T is not a surjective linear transformation
• B. There exists k ∈ \(\N\) such that T^k is the zero linear transformation
• C. 1 and 2 are the eigenvalues of T
• D. There exists r ∈ \(\N\) such that (T - I)r is the zero linear transformation, where I is the identity map on P₇(x)

Answer 1

The Correct Option is D

The linear transformation \( T(f(x)) = f(x) + \frac{df(x)}{dx} \) involves adding the function \( f(x) \) to its derivative. We need to examine the properties of \( T \): - For statement (A): Since \( T \) involves both the function and its derivative, it is not surjective. It cannot produce every polynomial because the derivative of a polynomial reduces the degree of the polynomial by 1, and \( T \) does not cover all polynomials in \( P_7(x) \). Hence, \( T \) is not surjective. Therefore, statement (A) is incorrect. - For statement (B): We know that applying the derivative repeatedly eventually results in the zero polynomial. After \( 8 \) derivatives (because the degree of the polynomial is at most 7), all terms of the polynomial vanish. Therefore, \( T^8 \) would be the zero transformation, but there is no such \( k \) less than 8. Hence, statement (B) is not true. - For statement (C): The eigenvalues of \( T \) are not necessarily \( 1 \) and \( 2 \). To find the eigenvalues, we need to solve \( T(f(x)) = \lambda f(x) \), and this involves more detailed work. Hence, statement (C) is not true. - For statement (D): Since the transformation involves both a function and its derivative, the operator \( T - I \) (where \( I \) is the identity) will eventually reach the zero transformation after a finite number of applications, as derivatives of polynomials of degree at most 7 will eventually vanish. Specifically, \( (T - I)^8 = 0 \). This makes statement (D) true. Thus, the correct answer is (D): There exists \( r \in \mathbb{N} \) such that \( (T - I)^r \) is the zero linear transformation, where \( I \) is the identity map on \( P_7(x) \).