91Ó°ÊÓ

A linear transformation is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. This means if you have a transformation \( T \), then for any vectors \( u \) and \( v \), and a scalar \( c \), the following must hold:

• \( T(u + v) = T(u) + T(v) \)
• \( T(cu) = cT(u) \)

This ensures that the structure of the space is maintained under the transformation.
In this exercise, the linear transformation \( T \) is defined from the polynomial space of degree 2, \( \mathscr{P}_{2} \), to the real numbers \( \mathbb{R} \). It acts on a polynomial by taking its derivative and evaluating it at zero:
\[ T(p(x)) = p'(0) \]This transformation takes a quadratic polynomial and produces a single real number, reflecting the slope of the tangent line at \( x = 0 \).

The kernel of a linear transformation is the set of all inputs that are mapped to the zero vector in the codomain. For a transformation \( T: V \rightarrow W \), a vector \( v \) is in the \( \text{Kernel}(T) \) if \( T(v) = 0 \).
In this exercise, the kernel consists of all polynomials whose derivative evaluated at 0 is zero.
Consider a polynomial \( p(x) = ax^2 + bx + c \). Its derivative is \( p'(x) = 2ax + b \). The condition for \( T(p(x)) = 0 \) is \( p'(0) = 0 \), which translates to \( b = 0 \).
This means the kernel includes polynomials of the form \( ax^2 + c \). These polynomials span a subspace with a basis \( \{ x^2, 1 \} \), and the dimension (or nullity) of the kernel is 2.

A polynomial space \( \mathscr{P}_{n} \) is the set of all polynomials of degree \( n \) or less. These polynomials can be expressed in the general form \( a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \), where the \( a_i \)'s are real numbers.
In our exercise, we deal with \( \mathscr{P}_{2} \), which consists of all polynomials of degree 2 or lower. Thus it includes polynomials like \( ax^2 + bx + c \), providing a three-dimensional space with the standard basis \( \{ 1, x, x^2 \} \).
The concept of polynomial space is fundamental to understanding transformations like \( T \), as it defines the domain over which such transformations operate. Here, the transformation maps this space to \( \mathbb{R} \), streamlining complex polynomial functions into simple real numbers via derivatives.

The derivative of a polynomial is an operation that gives the slope of the polynomial's graph at any given point. For any polynomial \( p(x) = ax^2 + bx + c \), its derivative is calculated as:

• \( p'(x) = 2ax + b \)

This derivative expresses the rate of change of the polynomial. Evaluating the derivative at a particular point—such as \( x = 0 \) in this exercise—provides the slope at that point.
In the context of this linear transformation, \( T(p(x)) = p'(0) \), we are particularly interested in the value of the derivative at zero. This allows us to determine which polynomials in \( \mathscr{P}_{2} \) will map to zero, an essential aspect of finding the kernel.