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 Tk 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

Answer: (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 2

The correct option is (D) : There exists r ∈ $\N$ such that (T - I)r is the zero linear transformation, where I is the identity map on P7(x).