Problem 15 - Differential Equations and Linear Algebra

Question

Determine a formula for the linear transformation meeting the given conditions. \(T: P_{2}(\mathbb{R}) \rightarrow M_{2}(\mathbb{R})\) such that $$T\left(x^{2}-x-3\right)=\left[\begin{array}{rr} -2 & 1 \\ -4 & -1 \end{array}\right], T(2 x+5)=\left[\begin{array}{rr} 0 & 1 \\ 2 & -2 \end{array}\right]$$ and $$T(6)=\left[\begin{array}{cc} 12 & 6 \\ 6 & 18 \end{array}\right]$$

Step-by-Step Solution

Answer

The general formula for the linear transformation T, for any quadratic polynomial p(x) with coefficients a, b, and c is: $$T[p(x)] = \left[\begin{array}{rr} -2a & a+b \\ -4a+2b+6c & -a-2b+18c \end{array}\right]$$

Step 1: Let a general quadratic polynomial \(p(x) \in P_2(\mathbb{R})\) be expressed as: $$p(x) = ax^2 + bx + c$$ Our goal is to express p(x) in terms of the given polynomial inputs \(x^2-x-3\), \(2x+5\), and \(6\). #step2-Find coefficients of the linear combination#

To express p(x) in terms of the given polynomial inputs, we need to find the coefficients \(\alpha, \beta, \gamma\) such that: $$p(x) = \alpha(x^2-x-3) + \beta(2x+5) + \gamma(6)$$ Then we can rewrite p(x) as: $$p(x) = \alpha x^2 -\alpha x -3\alpha + 2\beta x + 5\beta + 6\gamma$$ #step3-Group coefficients#

Step 2: Now, regroup the coefficients of the same powers of x in the previous equation: $$p(x) = (\alpha) x^2 + (2\beta - \alpha) x + (-3\alpha + 5\beta + 6\gamma)$$ #step4-Write as a system of equations#

Equate the coefficients of p(x) in the initial and final expressions, which leads us to the following system of equations: $$\begin{cases} \alpha = a \\ 2\beta - \alpha = b \\ -3\alpha + 5\beta + 6\gamma = c \end{cases}$$ #step5-Solve the system of equations#

Step 3: We can now solve this system to find \(\alpha, \beta, \gamma\) in terms of a, b, and c as follows: Find \(\beta\) from the second equation: $$\beta = \frac{b+\alpha}{2} = \frac{b+a}{2}$$ Substitute this value for \(\beta\) into the third equation and find \(\gamma\): $$\gamma = \frac{5b - 9a + 18c}{36}$$ #step6-Apply linearity of T#

Now that we have found the coefficients of the linear combination, we can apply the linearity of T. Since T is a linear transformation, it follows the property: $$T[p(x)] = T [\alpha(x^2-x-3)+\beta(2x+5)+\gamma(6)]$$ $$T[p(x)] = \alpha T[x^2-x-3]+\beta T[2x+5]+ \gamma T[6]$$ #step7-Substitute given values of T#

Step 4: Substitute the given values for T applied to the polynomial inputs to get the formula for the linear transformation T: $$T[p(x)] = \alpha \left[\begin{array}{rr} -2 & 1 \\ -4 & -1 \end{array}\right] + \beta \left[\begin{array}{rr} 0 & 1 \\ 2 & -2 \end{array}\right] + \gamma \left[\begin{array}{rr} 12 & 6 \\ 6 & 18 \end{array}\right]$$ #step8-Find T[p(x)]#

Finally, we have the general formula for the linear transformation T, for any quadratic polynomial p(x) with coefficients a, b, and c: $$T[p(x)] = \left[\begin{array}{rr} -2a & a+b \\ -4a+2b+6c & -a-2b+18c \end{array}\right]$$

Key Concepts

Quadratic Polynomials

A quadratic polynomial is an algebraic expression of degree two. It generally takes the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. Quadratic polynomials are fundamental building blocks in algebra and have many applications, such as describing parabolic trajectories and optimization problems.

When we work with quadratic polynomials in the context of linear transformations, it is essential to express them in a form that allows us to understand and manipulate them easily. In our specific exercise, the aim was to express a general quadratic polynomial \( p(x) \) as a linear combination of given polynomial inputs: \( x^2-x-3 \), \( 2x+5 \), and \( 6 \).

This technique of expressing one polynomial using others simplifies computations and makes it possible to apply transformations consistently. By breaking polynomials into known forms, we can explore their inherent properties and behavior under transformations like linear mappings.

Matrix Representation

Matrix representation is a powerful tool in linear algebra that allows us to express linear transformations using matrices. By representing transformations as matrices, we can handle complex operations with more simplicity and precision. In the exercise, the linear transformation \( T \) maps quadratic polynomials to 2x2 matrices, which is signified as \( T: P_2(\mathbb{R}) \rightarrow M_2(\mathbb{R}) \).

Understanding how to use matrix representation involves recognizing how each component of the polynomial corresponds to elements within a matrix. For instance, the transformation \( T \) in our case takes polynomials such as \( x^2-x-3 \) and returns matrices like \( \begin{bmatrix} -2 & 1 \ -4 & -1 \end{bmatrix} \). This link between polynomials and matrices allows us to apply linear algebra techniques to solve and interpret various mathematical problems.

Matrices organize information systematically, and when we dive into solving systems of equations involving transformations, it’s this organizational ability that provides clarity. Through matrix operations, addition, multiplication, and scalar multiplication become straightforward tasks.

System of Equations

A system of equations is a collection of two or more equations involving the same set of variables. Solving a system of equations means finding the values of the variables that satisfy all the equations simultaneously. In our exercise, we derive a system from equating coefficients of a polynomial with its representation in terms of given inputs.

The system of equations we developed was:

• \( \alpha = a \)
• \( 2\beta - \alpha = b \)
• \( -3\alpha + 5\beta + 6\gamma = c \)

Simplifying these equations helps us determine the values of \( \alpha \), \( \beta \), and \( \gamma \) in terms of \( a \), \( b \), and \( c \). This process ultimately allows us to express polynomial transformations more fully and apply them decisively.

Systems of equations often come up in many fields, whether you're determining electrical currents in circuits or finding unknown forces in physics. Mastery of these systems is vital for solving real-world and theoretical problems where multiple conditions must be met at once.