91Ó°ÊÓ

In the realm of linear transformations like our function \( T \), understanding "one-to-one mapping" is essential. A transformation is considered one-to-one, or injective, if it assigns distinct outputs to distinct inputs. In simpler terms, if \( T(p(x)) = T(q(x)) \) implies that \( p(x) = q(x) \), then \( T \) is one-to-one.
In our exercise, we set the outputs equal, i.e., \( T(p(x)) = T(q(x)) \), and ended up with equations requiring \( c_p = c_q \) and \( a_p + b_p + c_p = a_q + b_q + c_q \). Even after simplifying, we found infinite possibilities of \( a_p \), \( b_p \), \( a_q \), and \( b_q \) such that the polynomials can still be distinct. This leads to the conclusion that \( T \) isn't one-to-one as different inputs can yield the same result.

Onto mapping, or surjectivity, is about ensuring the function covers every possible output in its target space. A transformation \( T \) is onto if, for every vector in the target space, there is a polynomial that maps to it.
In our case, we consider any vector \( \begin{bmatrix} y_1 \ y_2 \end{bmatrix} \) in \( \mathbb{R}^2 \) and pose the question: is there a polynomial \( p(x) = ax^2 + bx + c \) such that \( T(p(x)) = \begin{bmatrix} y_1 \ y_2 \end{bmatrix} \)? We see through the equations \( c = y_1 \) and \( a + b + c = y_2 \) that solutions can be found for any choice of \( y_1 \) and \( y_2 \). Thus, \( T \) is onto, demonstrating that we can reach any point in \( \mathbb{R}^2 \) using our polynomials.

Polynomial functions form the backbone of the transformation in our example. Simply put, a polynomial is a mathematical expression involving sums and powers of variables, typically \( x \), with coefficients. In \( \mathscr{P}_2 \), we specifically look at quadratic polynomials \( p(x) = ax^2 + bx + c \).
The transformation \( T \) cleverly uses these polynomials: it evaluates them at specific points—in this case \( x = 0 \) and \( x = 1 \)—to form a vector in \( \mathbb{R}^2 \). With polynomials of degree up to 2, their behavior and structure allow us to explore concepts like one-to-one and onto mappings, as seen in our example. By examining outputs at particular points, we capture essential information about the polynomial's properties.

In analyzing linear transformations, systems of equations become crucial tools. They help us decode relationships between inputs and outputs of the transformation. Here, we encountered two specific equations: \( c_p = c_q \) and \( a_p + b_p + c_p = a_q + b_q + c_q \).
By resolving these, we checked the injectivity of \( T \)—whether different inputs could lead to different outputs. The system showed multiple solutions for \( a \) and \( b \), signaling that \( T \) isn't one-to-one. Similarly, the system \( c = y_1 \) and \( a + b + c = y_2 \) confirmed the surjectivity, or onto nature, of \( T \), revealing that all output vectors in \( \mathbb{R}^2 \) have a corresponding input. Understanding these systems is key to unpacking the behavior of transformations.