91Ó°ÊÓ

When dealing with linear transformations, especially in linear algebra, a standard matrix is one of the most important tools. It provides a convenient way to represent a linear transformation applied to a basis of a vector space.

For the transformation \( T: \mathscr{P}_2 \to \mathscr{P}_2 \) defined as \( T(p(x))=p(x)+p'(x) \), we consider polynomials up to the second degree. This leads to forming a standard matrix that represents this transformation.

The general polynomial \( p(x) = a_0 + a_1 x + a_2 x^2 \) transforms under \( T \) to \( a_0 + (2a_1 + 2a_2)x + a_2 x^2 \). By comparing coefficients, we see how each component of the polynomial is transformed.

The corresponding standard matrix is:

\[A = \begin{pmatrix} 1 & 0 & 0 \ 0 & 2 & 0 \ 0 & 1 & 1 \end{pmatrix}.\]

Understanding this matrix allows us to work with polynomials in a structured, linear fashion, providing the groundwork to examine properties like invertibility.

The determinant is a scalar value that encodes certain properties of a matrix, which helps in understanding the invertibility of linear transformations. For a square matrix, if the determinant is zero, the matrix is said to be singular, meaning it does not have an inverse.

In our exercise, for matrix \( A \):

\[\text{det}(A) = \begin{vmatrix} 1 & 0 & 0 \ 0 & 2 & 0 \ 0 & 1 & 1 \end{vmatrix} = 1(2 \times 1 - 0) - 0 + 0 = 2.\]

The determinant here is 2, which indicates that the matrix is invertible. This means the linear transformation \( T \) is also invertible. The non-zero determinant directly leads us to explore the matrix inverse, which is essential for determining the inverse of the transformation.

Polynomial transformation involves transforming a polynomial function into another through operations like addition, scaling, differentiation, or integration. In this exercise, the transformation \( T \) takes a polynomial \( p(x) \), adds its derivative \( p'(x) \), resulting in a new polynomial.

Given the polynomial \( p(x) = a_0 + a_1 x + a_2 x^2 \), its derivative is \( p'(x) = a_1 + 2a_2 x \). The transformation \( T(p(x)) = p(x) + p'(x) \) yields:

• The constant term \( a_0 \) remains unchanged.
• The coefficient for \( x^1 \) becomes \( 2a_1 + 2a_2 \).
• The coefficient for \( x^2 \) stays \( a_2 \).

This transformation is linear since it satisfies linearity conditions, making it a crucial aspect of algebraic operations on polynomial functions.

The matrix inverse is a central concept when discussing invertibility. For any square matrix \( A \), finding its inverse \( A^{-1} \) enables the reverse transformation of any linear system it governs.

If a matrix is invertible, meaning its determinant is non-zero, it has an inverse. For the specified matrix \( A \):

\[A^{-1} = \begin{pmatrix} 1 & 0 & 0 \ 0 & 0.5 & 0 \ 0 & -0.5 & 1 \end{pmatrix}.\]

This inverse matrix is used to define the inverse of the transformation \( T \), symbolized as \( T^{-1} \). By applying \( A^{-1} \) to the transformed polynomial, we retrieve the original polynomial components.

For polynomial \( q(x) = b_0 + b_1 x + b_2 x^2 \), the inverse transformation gives:

• The constant term remains \( b_0 \).
• The coefficient for \( x^1 \) is adjusted to \( 0.5 b_1 \).
• For \( x^2 \), the coefficient becomes \( b_2 - 0.5b_1 \).

This process highlights how each polynomial term is systematically reversed to derive the original input using the inverse matrix.