Problem 19 - 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 \(V.\) $$\begin{array}{l}V=\mathbb{R}^{3} ; B=\\{(2,-5,0),(3,0,5),(8,-2,-9)\\} \\\C=\\{(1,-1,1),(2,0,1),(0,1,3)\\} \end{array}.$$

Step-by-Step Solution

Answer

The change-of-basis matrix from ordered basis B to ordered basis C, denoted as \(P_{C \leftarrow B}\), is found to be: $$ P_{C \leftarrow B} = \begin{bmatrix} 2 & 2 & 8 \\ 1 & 1 & 1 \\ 1 & -1 & -3 \end{bmatrix} $$

Step 1: Write down the given bases

The ordered bases B and C are given as follows: $$ B = \{(2,-5,0), (3,0,5), (8,-2,-9)\} \\ C = \{(1,-1,1), (2,0,1), (0,1,3)\} $$ Remember that we want to find the change-of-basis matrix from basis B to basis C, denoted as \(P_{C \leftarrow B}\).

Step 2: Express vectors in basis B as linear combinations of vectors in basis C

We want to find the coefficients \(c_1, c_2,\) and \(c_3\) for each vector in basis B, so that each vector in B can be represented as a linear combination of the vectors in basis C: $$ b_i = c_1c_i^1 + c_2c_i^2 + c_3c_i^3, $$ where \(b_i\) are the vectors of basis B and \(c_i^j\) are the vectors of basis C. This can be written as a system of linear equations in matrix form: $$ \begin{bmatrix} 1 & 2 & 0 \\ -1 & 0 & 1 \\ 1 & 1 & 3 \end{bmatrix} \begin{bmatrix} c_1 \\ c_2 \\ c_3 \end{bmatrix} = \begin{bmatrix} b_{i1} \\ b_{i2} \\ b_{i3} \end{bmatrix} $$ Solving this system for each vector in basis B will give us the coefficients for the linear combinations.

Step 3: Solve the systems of linear equations for each vector in basis B

Solving the above matrix equation for each vector in basis B, we find the following coefficients: For \(b_1 = (2,-5,0)\) $$ \begin{bmatrix} c_1 \\ c_2 \\ c_3 \end{bmatrix} = \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix} $$ For \(b_2 = (3,0,5)\) $$ \begin{bmatrix} c_1 \\ c_2 \\ c_3 \end{bmatrix} = \begin{bmatrix} 2 \\ 1 \\ -1 \end{bmatrix} $$ For \(b_3 = (8,-2,-9)\): $$ \begin{bmatrix} c_1 \\ c_2 \\ c_3 \end{bmatrix} = \begin{bmatrix} 8 \\ 1 \\ -3 \end{bmatrix} $$

Step 4: Form the change-of-basis matrix P

Now that we found the coefficients for the linear combinations of the vectors in basis B, we can form the change-of-basis matrix \(P_{C \leftarrow B}\) by putting the coefficients found in step 3 as columns: $$ P_{C \leftarrow B} = \begin{bmatrix} 2 & 2 & 8 \\ 1 & 1 & 1 \\ 1 & -1 & -3 \end{bmatrix} $$ This is the change-of-basis matrix from ordered basis B to ordered basis C.

Key Concepts

Linear Combinations

Understanding linear combinations is crucial when working with vector spaces and change of basis. A linear combination is a way of expressing a vector as a sum of scalar multiples of other vectors.

• For example, if you have a set of vectors, you can create new vectors by combining these using specific weights or scalars.
• In the context of the change-of-basis matrix, linear combinations allow us to express the vectors from one basis in terms of the vectors of another basis.

To break it down with some familiar notation: suppose you have vectors \(v_1, v_2,...,v_n\) and you want to express another vector \(v\) as a linear combination of those vectors. In mathematical terms, this means writing \(v = a_1v_1 + a_2v_2 + ... + a_nv_n\) where the \(a_i\)'s are real numbers.
The goal in this exercise is to find scalars \(c_1, c_2, c_3\) so that each vector of the base \(B\) can be expressed as a linear combination of vectors in the base \(C\). This transforms our understanding of the original vectors in terms of new ones, facilitating transformations like change of basis.

Vector Spaces

Vector spaces offer a fundamental backdrop to many areas in mathematics and applied sciences. They are essentially a collection of vectors where you can perform vector addition and scalar multiplication.

• These operations adhere to certain rules, like associativity, commutativity, and distributive properties.
• Each vector space is also linear over a field, which means the scalars that multiply the vectors come from a specified field like real numbers.

In the context of basis change, particularly with \(\mathbb{R}^3\) in our exercise, a vector space encompasses all possible combinations of the given basis vectors. The basis \(B\) and \(C\) are two different ways to "span" this space, meaning both can be used to represent any vector in the space though the components would differ.
This highlights one key property of vector spaces: Any vector in the space can be expressed uniquely as a linear combination of the basis vectors. Changing the basis is akin to choosing a new set of directions, which can be described by a change-of-basis matrix, mapping one basis to another.

Matrix Equations

Transforming between bases in vector spaces relies heavily on solving matrix equations. These equations involve vectors from one basis and expressing them in terms of another using matrices.

• Each vector equation from the exercise becomes a matrix equation, which follows the form \(A \mathbf{x} = \mathbf{b}\).
• Here, \(A\) represents the matrix formed from the coefficients of the target basis vectors, \(\mathbf{x}\) is the vector of unknowns (the scalars), and \(\mathbf{b}\) is the vector from the original basis.

Understanding how to solve these systems is essential for deriving the change-of-basis matrix. In the exercise, we constructed rows and columns corresponding to each vector from basis \(B\) and used matrix operations to find the required scalars.
By doing so, we formed the matrix \(P_{C \leftarrow B}\), whose columns correspond to the coefficients needed to express each vector in basis \(B\) in terms of basis \(C\). Efficient matrix manipulation is the key technique here, underscoring the importance of matrix algebra in linear transformations.