Question

Let \( T, S : P_4(\mathbb{R}) \to P_4(\mathbb{R}) \) be the linear transformations defined by \[ T(p(x)) = xp'(x), \quad S(p(x)) = (x + 1)p'(x) \] for all \( p(x) \in P_4(\mathbb{R}) \). Then, the nullity of the composition \( S \circ T \) is ................

Hint:

To find the nullity of a composition of linear transformations, first express the composition explicitly and then solve for the kernel. The nullity is the dimension of the kernel.

Answer 1

Step 1: Analyze the transformations.
The transformation \( T \) is defined by \( T(p(x)) = xp'(x) \), which involves multiplying the derivative of \( p(x) \) by \( x \). The transformation \( S \) is defined by \( S(p(x)) = (x + 1)p'(x) \), which involves multiplying the derivative of \( p(x) \) by \( (x + 1) \). Step 2: Composition of \( S \circ T \).
The composition \( S \circ T \) is given by: \[ (S \circ T)(p(x)) = S(T(p(x))) = (x + 1) \cdot (xp'(x))' = (x + 1) \cdot (p'(x) + xp''(x)). \] Step 3: Nullity of the composition.
To find the nullity, we need to find the dimension of the kernel of \( S \circ T \). We check which polynomials \( p(x) \in P_4(\mathbb{R}) \) satisfy \( (S \circ T)(p(x)) = 0 \). This is equivalent to solving \( (x + 1)(p'(x) + xp''(x)) = 0 \). The solutions will give the null space of \( S \circ T \), which has dimension 1. Final Answer: \[ \boxed{1}. \]