91Ó°ÊÓ

The kernel of a linear transformation is a set that contains all the elements from the domain that are mapped to the zero vector in the codomain. In our exercise, we have a transformation, \( T: \mathscr{P}_{2} \to \mathbb{R}^{2} \), that maps polynomials \( a + bx + cx^2 \) to vectors \( \begin{bmatrix} a - b \ b + c \end{bmatrix} \). To find \( \text{ker}(T) \), we need the transformation output to be the zero vector, \( \begin{bmatrix} 0 \ 0 \end{bmatrix} \).This requirement results in a system of linear equations:

• \( a - b = 0 \)
• \( b + c = 0 \)

By solving these equations, we find \( a = b \) and \( c = -b \). Thus, polynomials in the kernel are of the form \( b(1 + x - x^2) \), constituting a one-dimensional subspace of \( \mathscr{P}_{2} \). This means any multiple of \( 1 + x - x^2 \) will lie in the kernel.

The range of a linear transformation, also known as the image, is the set of all possible outputs. For transformation \( T: \mathscr{P}_{2} \to \mathbb{R}^{2} \), the task is to uncover all vectors of the form \( \begin{bmatrix} a-b \ b+c \end{bmatrix} \) that can be derived from polynomials \( a + bx + cx^2 \).By selecting different values for \( a \), \( b \), and \( c \), we generate varied vectors. For example:

• Selecting \( a = 0, b = 0, c = 0 \) yields \( \begin{bmatrix} 0 \ 0 \end{bmatrix} \).
• Choosing \( a = 1, b = 1, c = 0 \) results in \( \begin{bmatrix} 0 \ 1 \end{bmatrix} \).
• With \( a = 1, b = 0, c = -1 \), we get \( \begin{bmatrix} 1 \ -1 \end{bmatrix} \).

Hence, any vector \( \begin{bmatrix} a - b \ b + c \end{bmatrix} \) is achievable, essentially spanning the entire \( \mathbb{R}^2 \). Thus, \( \text{range}(T) \) is all of \( \mathbb{R}^2 \).

A polynomial space is a vector space whose elements are polynomials. In this context, we consider \( \mathscr{P}_{2} \), which is the space of all polynomials of degree 2 or less. Examples of elements in this space include linear terms like \( b + cx \) and quadratic terms like \( cx^2 \).For our linear transformation, \( T: \mathscr{P}_{2} \to \mathbb{R}^{2} \), each input is a polynomial that can be expressed in terms of \( a + bx + cx^2 \). The coefficients \( a \), \( b \), and \( c \) act as coordinates in this polynomial space. Operations like addition and scalar multiplication adhere to the rules of vector spaces, maintaining closure, associativity, and distributivity.Polynomial spaces are crucial for capturing linear relationships and transformations in mathematical contexts, making them a powerful tool in analyzing and solving algebraic problems.

A vector space is a fundamental concept in linear algebra. It's defined as a collection of vectors that can undergo addition and scalar multiplication while satisfying certain axioms. In our exercise, transformations map polynomials of \( \mathscr{P}_{2} \) to vectors in \( \mathbb{R}^{2} \).Key properties of vector spaces include:

• Closure under addition: Sum of two vectors results in a vector within the space.
• Closure under scalar multiplication: Multiplying a vector by a scalar produces a vector within the space.
• Presence of zero vector: A vector exists that, when added to any vector, returns the same vector.
• Existence of additive inverses: Each vector has a corresponding negative vector that sums to the zero vector.

The understanding of vector spaces allows students to handle a wide array of algebraic and geometric problems, providing a vital framework for tackling complex transformations like those examined in this exercise.

Systems of linear equations are critical tools for analyzing linear transformations. They involve multiple linear equations that share common variables. In our exercise's context, when finding \( \text{ker}(T) \), the system is:

• \( a - b = 0 \)
• \( b + c = 0 \)

Solving this system uncovers relationships between variables, essential for determining the kernel of \( T \). By finding variable values that satisfy all equations simultaneously, we pinpoint the element structure of \( \text{ker}(T) \).Systems of equations represent interconnected relationships and are pivotal in unveiling the potential solutions within a given mathematical framework. By breaking down complex transformations into systems, students develop a systematic approach to solve algebraic challenges.