Problem 20 - Differential Equations and Linear Algebra

Question

Let \(V\) denote the vector space of \(2 \times 2\) symmetric matrices and define \(T: V \rightarrow P_{2}(\mathbb{R})\) by $$T\left(\left[\begin{array}{ll} a & b \\ b & c \end{array}\right]\right)=a x^{2}+b x+c$$ Determine whether \(T\) is one-to-one, onto, both, or neither. Find \(T^{-1}\) or explain why it does not exist.

Step-by-Step Solution

Answer

The transformation \(T\) is both one-to-one and onto. The inverse transformation \(T^{-1}\) exists, and is given by \(T^{-1}(px^2 + qx + r) = \begin{bmatrix} p & q \\ q & r\end{bmatrix}\).

Step 1: Analysis of the Transformation

Given the transformation \[ T: \begin{bmatrix} a & b \\ b & c \end{bmatrix} \rightarrow ax^2 +bx +c\] it's easy to see that each matrix \(\begin{bmatrix} a & b \\ b & c \end{bmatrix}\) will correspond to a unique polynomial \(ax^2 +bx +c\). Hence, the transformation is one-to-one.

Step 2: Verification of One-to-One

To formally prove that \(T\) is one-to-one, suppose \(T(M_1) = T(M_2)\), where \(M_1\) and \(M_2\) are matrices in \(V\). We need to show that this implies \(M_1 = M_2\). Let \(M_1 = \begin{bmatrix} a_1 & b_1 \\ b_1 & c_1\end{bmatrix}\) and \(M_2 = \begin{bmatrix} a_2 & b_2 \\ b_2 & c_2 \end{bmatrix}\) If \(T(M_1) = T(M_2)\) then \(a_1x^2 +b_1x +c_1 = a_2x^2 +b_2x +c_2 \) Equating coefficients, we have that \(a_1 = a_2\), \(b_1 = b_2\), and \(c_1 = c_2\) Therefore \(M_1 = M_2\) So, the transformation \(T\) is one-to-one.

Step 3: Verification of Onto

To check whether the transformation \(T\) is onto, consider a generic polynomial of degree 2 or less: \(p(x) = dx^2 +ex +f\). We can clearly find a matrix \(M = \begin{bmatrix} d & e \\ e & f\end{bmatrix}\) such that \(T(M) = p(x)\). So, every polynomial of degree 2 or less is the image of some matrix \(M\) under the transformation \(T\). This means the transformation \(T\) is onto.

Step 4: The Inverse Transformation

Since \(T\) is both one-to-one and onto, it has an inverse \(T^{-1}\). This inverse transformation \(T^{-1}\) would act on a polynomial \(px^2 +qx +r\) of degree 2 or less, and it would map this polynomial back to the original 2x2 symmetric matrix. From observation we know that \(T^{-1}(px^2 +qx +r) = \begin{bmatrix} p & q \\ q & r\end{bmatrix}\). So, \(T\) is one-to-one, onto, and its inverse \(T^{-1}\) exists.

Key Concepts

Vector Space

A vector space is a fundamental structure in linear algebra. It consists of a set of objects, called vectors, along with operations for addition and scalar multiplication. These operations must satisfy certain axioms, such as associativity, distributivity, and the existence of an additive identity (zero vector). For example, the vector space of symmetric matrices of size \(2 \times 2\) includes all matrices of the form \(\begin{bmatrix} a & b \ b & c \end{bmatrix}\).

Symmetric matrices themselves form a vector space because:

• The sum of any two symmetric matrices is again a symmetric matrix.
• Scalar multiples of symmetric matrices are also symmetric.
• There exists a zero matrix (\(\begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix}\)) in the space.

In the context of our exercise, we are working within the vector space \(V\) of \(2 \times 2\) symmetric matrices. This space is important because it defines the domain of the transformation \(T\) that maps these matrices to polynomials of degree 2 or less. Understanding vector spaces allows us to grasp how transformations like \(T\) are structured and which properties they have, such as being one-to-one and onto.

Symmetric Matrices

Symmetric matrices have a beautiful property: they are equal to their transpose. For a \(2 \times 2\) symmetric matrix \(\begin{bmatrix} a & b \ b & c \end{bmatrix}\), this means \(a\) and \(c\) are on the diagonal while \(b\) occupies both off-diagonal elements. Such matrices are important in many applications, including physics and statistics.

Properties of symmetric matrices:

• They are a subset of square matrices, meaning they have equal numbers of rows and columns.
• Symmetric matrices are always diagonalizable, which means they can be broken down into eigenvalues and eigenvectors.
• Matrix operations, such as addition and multiplication by scalars, keep matrices symmetric.

In the given transformation \(T\), these matrices are transformed into polynomials where the positions \(a\), \(b\), and \(c\) directly map to the coefficients of \(x^2\), \(x\), and the constant term, respectively. This illustrates how each unique symmetric matrix maps to a distinct polynomial, ensuring our transformation is one-to-one and onto.

Polynomials

Polynomials are elementary functions formed with coefficients that make up terms involving powers of \(x\). For example, a polynomial \(ax^2 + bx + c\) is quadratic, meaning it has a degree of 2. This ties directly to our transformation \(T\) because each symmetric matrix is mapped to a polynomial of degree 2 or less.

Some key aspects of polynomials:

• The degree of a polynomial is determined by the highest power of \(x\) with a non-zero coefficient.
• Each polynomial corresponds to a unique function, which can be graphed as a curve.
• Operations such as addition, subtraction, and multiplication can be performed on polynomials, which play a crucial role in their applications.

In our previous exercise, the transformation \(T\) converts symmetric matrices into polynomials, creating straightforward polynomial expressions like \(ax^2 + bx + c\). Understanding how polynomials are structured reinforces why our transformation is one-to-one (each matrix corresponds to a unique polynomial) and onto (any polynomial of compatible form can be represented by such a matrix). This makes it possible for an inverse transformation \(T^{-1}\) to exist, effectively reversing this mapping from polynomials back to matrices.