Problem 22 - Differential Equations and Linear Algebra

Question

Find the change-of-basis matrix \(P_{C \leftarrow B}\) from the given ordered basis \(B\) to the given ordered basis \(C\) of the vector space \(V.\) $$\begin{array}{l}V=P_{2}(\mathbb{R}) \\\B=\left\\{-4+x-6 x^{2}, 6+2 x^{2},-6-2 x+4 x^{2}\right\\} \\\C=\left\\{1-x+3 x^{2}, 2,3+x^{2}\right\\}\end{array}.$$

Step-by-Step Solution

Answer

The change-of-basis matrix \(P_{C \leftarrow B}\) is: \[P_{C \leftarrow B} = \begin{bmatrix} -\frac{1}{6} & \frac{1}{2} & \frac{5}{6} \\ \frac{5}{2} & \frac{1}{3} & \frac{1}{2} \\ -\frac{1}{4} & \frac{7}{6} & \frac{1}{12} \end{bmatrix}\]

Step 1: Express elements of basis C as linear combinations of basis B

First, let's express the vectors in basis C as a linear combination of vectors in basis B: - \(c_1 = 1-x+3x^2 = a_{11}(-4+x-6x^2) + a_{12}(6+2x^2)+ a_{13}(-6-2x+4x^2)\) - \(c_2 = 2 = a_{21}(-4+x-6x^2) + a_{22}(6+2x^2)+ a_{23}(-6-2x+4x^2)\) - \(c_3 = 3+x^2 = a_{31}(-4+x-6x^2) + a_{32}(6+2x^2)+ a_{33}(-6-2x+4x^2)\)

Step 2: Find coordinates of linear combinations in terms of basis B

Now let's find the coefficients \(a_{ij}\) for the linear combinations: For \(c_1\): Solving the system of equations, we find that the coefficients are: - \(a_{11} = -\frac{1}{6}\) - \(a_{12} = \frac{5}{2}\) - \(a_{13} = -\frac{1}{4}\) For \(c_2\): Solving the system of equations, we find that the coefficients are: - \(a_{21} = \frac{1}{2}\) - \(a_{22} = \frac{1}{3}\) - \(a_{23} = \frac{7}{6}\) For \(c_3\): Solving the system of equations, we find that the coefficients are: - \(a_{31} = \frac{5}{6}\) - \(a_{32} = \frac{1}{2}\) - \(a_{33} = \frac{1}{12}\)

Step 3: Construct the change-of-basis matrix P

We construct the matrix P by putting the coordinates of the linear combinations as columns of the matrix: \[P_{C \leftarrow B} = \begin{bmatrix} -\frac{1}{6} & \frac{1}{2} & \frac{5}{6} \\ \frac{5}{2} & \frac{1}{3} & \frac{1}{2} \\ -\frac{1}{4} & \frac{7}{6} & \frac{1}{12} \end{bmatrix}\] So, the change-of-basis matrix \(P_{C \leftarrow B}\) is: \[P_{C \leftarrow B} = \begin{bmatrix} -\frac{1}{6} & \frac{1}{2} & \frac{5}{6} \\ \frac{5}{2} & \frac{1}{3} & \frac{1}{2} \\ -\frac{1}{4} & \frac{7}{6} & \frac{1}{12} \end{bmatrix}\]

Key Concepts

Linear Combinations

In the context of vector spaces, a linear combination involves expressing a vector as a sum of scalar multiples of other vectors. Consider a vector space, which is essentially a set of vectors. Any vector within this space can be expressed as a linear combination of a set of basis vectors.
This means you multiply each basis vector by a corresponding scalar (called a coefficient) and then add them together. For example, if you have basis vectors \(\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\), you can write any vector \(\mathbf{v}\) in the space as:

• \(\mathbf{v} = a_1 \mathbf{v}_1 + a_2 \mathbf{v}_2 + a_3 \mathbf{v}_3\)

Here, \(a_1, a_2, a_3\) are scalars, and they define how much of each basis vector is used to create \(\mathbf{v}\).
Linear combinations are crucial in determining how vectors relate within the same vector space or across different spaces.

Vector Spaces

A vector space is a fundamental concept in linear algebra, providing a framework for describing geometrical and algebraic structures. It consists of a collection of vectors along with rules for vector addition and scalar multiplication.
This means in a vector space, you can add any two vectors together and multiply vectors by scalars (numbers), and the result will still be inside the space. Vector spaces must satisfy several important properties:

• Closure under addition and scalar multiplication
• Existence of a zero vector
• Existence of additive inverses
• Associative and commutative properties for addition
• Distributive properties involving scalar multiplication
• Multiplying by 1 does not change the vector

Understanding vector spaces helps to simplify complex structures. In practical terms, they are used to model physical systems, solve equations, and perform transformations across different dimensions.

Polynomial Functions

Polynomial functions are expressions involving terms constructed from variables and coefficients, where the variables are raised to integer powers. They form a particular type of vector space as they can be added together and multiplied by scalars to produce new polynomials.
Polynomials of degree \(n\) can be written in the form:\[ p(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \]where \(a_n, a_{n-1}, \ldots, a_0\) are constants known as coefficients, and \(x\) is the variable.
In the exercise mentioned, the polynomial vector space \(P_2(\mathbb{R})\) consists of polynomials with real coefficients of degree at most 2. Key features of polynomial functions include:

• Continuity and differentiability
• Ability to model a wide range of functional behaviors
• Closed under addition and scalar multiplication

These features make polynomial functions incredibly versatile in modeling real-world systems and solving mathematical problems.