Question

Consider the vector space \( V \) over \( \mathbb{R} \) of polynomial functions of degree less than or equal to 3 defined on \( \mathbb{R} \). Let \( T : V \to V \) be defined by \( (T f)(x) = f(x) - x f'(x) \). Then the rank of \( T \) is

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

The rank of a linear transformation is the dimension of its image. When applying a linear transformation to polynomials, the degree of the resulting polynomial can help determine the rank.

Answer 1

The Correct Option is C

Step 1: Understanding the linear transformation. 
The linear transformation \( T \) acts on polynomial functions \( f(x) \) by subtracting \( x f'(x) \) from \( f(x) \). We need to determine the rank of \( T \), which is the dimension of the image of \( T \). 
 

Step 2: Analyzing the action of \( T \). 
Consider a general polynomial of degree at most 3: \( f(x) = ax^3 + bx^2 + cx + d \). Applying \( T \) to \( f(x) \), we get: \[ (T f)(x) = ax^3 + bx^2 + cx + d - x(3ax^2 + 2bx + c) \] Simplifying this expression, we see that the result is a polynomial of degree at most 3. This means the image of \( T \) is spanned by polynomials of degree 3 or less, so the rank of \( T \) is 3. 
 

Step 3: Conclusion. 
The correct answer is (C) 3.