Finding Coordinate Matrix Of X In R^n Relative To Basis B'

Introduction

In linear algebra, the concept of a coordinate matrix is fundamental for understanding how vectors are represented in different bases. When we express a vector in a non-standard basis, the coordinate matrix provides a way to map the vector from the standard basis to the new basis. This is particularly useful in various applications such as computer graphics, data analysis, and engineering. This article aims to provide a comprehensive guide on how to find the coordinate matrix of a vector x in Rⁿ relative to a given basis B'. We will delve into the underlying theory, provide a step-by-step approach, and illustrate the method with a detailed example.

Coordinate matrices serve as a bridge between different vector representations, allowing us to perform transformations and computations more efficiently. Understanding how to compute these matrices is crucial for anyone working with linear transformations and vector spaces. In this article, we will explore the process using a specific example where B' = {(8,11,0), (7,0,10), (1,4,6)} and x = (-11,19,-18). By breaking down the problem into manageable steps, we will demonstrate how to express x as a linear combination of the basis vectors in B', and subsequently, form the coordinate matrix. This article is designed to be accessible to both students and professionals seeking to solidify their understanding of linear algebra concepts.

By the end of this guide, you will have a clear understanding of the procedure for finding coordinate matrices and be able to apply this knowledge to solve similar problems. This skill is invaluable for anyone working in fields that rely on linear algebra, such as computer science, engineering, and mathematics. So, let's embark on this journey to unravel the intricacies of coordinate matrices and enhance your linear algebra toolkit.

Theoretical Background

Before diving into the computational aspects, it is essential to grasp the theoretical underpinnings of coordinate matrices. In essence, a coordinate matrix represents a vector in terms of a specific basis. Let's denote Rⁿ as an n-dimensional vector space, and B' = {b₁, b₂, ..., b_n} as a basis for this space. Any vector x in Rⁿ can be uniquely expressed as a linear combination of the basis vectors in B'. This means we can find scalars c₁, c₂, ..., c_n such that:

x = c₁b₁ + c₂b₂ + ... + c_nb_n

The scalars c₁, c₂, ..., c_n are called the coordinates of x relative to the basis B'. The coordinate matrix of x relative to B', denoted as [x]_B', is a column vector formed by these coordinates:

[x]_B' =

$$\begin{bmatrix} c_1 \\ c_2 \\ ... \\ c_n \end{bmatrix}$$

The key idea here is that the coordinate matrix provides a new representation of the vector x in the context of the basis B'. This representation is unique, given that B' is indeed a basis for Rⁿ. Understanding this uniqueness is crucial because it ensures that the transformation from x to [x]_B' is well-defined and reversible.

To find the coordinate matrix, we need to determine the scalars c₁, c₂, ..., c_n. This typically involves solving a system of linear equations. The process can be visualized as projecting the vector x onto the subspace spanned by the basis vectors in B'. The coefficients of these projections then form the entries of the coordinate matrix. The theoretical framework described here is not just an abstract mathematical concept; it has practical implications in numerous fields. For instance, in computer graphics, different bases can be used to represent objects in a scene, and the choice of basis can significantly impact the efficiency of rendering algorithms. Similarly, in data analysis, techniques like Principal Component Analysis (PCA) rely on changing bases to reduce the dimensionality of data while preserving essential information. Therefore, a solid understanding of coordinate matrices is invaluable for anyone working with vector spaces and linear transformations.

Step-by-Step Method to Find the Coordinate Matrix

Now that we have a solid theoretical foundation, let's outline the step-by-step method to find the coordinate matrix of a vector x in Rⁿ relative to a basis B'. This process involves setting up and solving a system of linear equations. Here’s how we approach it:

Step 1: Express x as a Linear Combination of Basis Vectors

The first step is to express the vector x as a linear combination of the basis vectors in B'. Given B' = {b₁, b₂, ..., b_n}, we need to find scalars c₁, c₂, ..., c_n such that:

x = c₁b₁ + c₂b₂ + ... + c_nb_n

This equation is the foundation of our approach. It states that x can be written as a weighted sum of the basis vectors, where the weights are the coordinates we are trying to find. The ability to express x in this form is guaranteed if B' is indeed a basis for Rⁿ.

Step 2: Set Up the System of Linear Equations

Once we have the linear combination, we can set up a system of linear equations. Each component of the vectors in the equation gives us one equation in the system. For example, if we are working in R³ and x = (x₁, x₂, x₃), b₁ = (b₁₁, b₁₂, b₁₃), b₂ = (b₂₁, b₂₂, b₂₃), and b₃ = (b₃₁, b₃₂, b₃₃), then the equation from Step 1 becomes:

(x₁, x₂, x₃) = c₁(b₁₁, b₁₂, b₁₃) + c₂(b₂₁, b₂₂, b₂₃) + c₃(b₃₁, b₃₂, b₃₃)

This vector equation can be broken down into the following system of linear equations:

• x₁ = c₁b₁₁ + c₂b₂₁ + c₃b₃₁
• x₂ = c₁b₁₂ + c₂b₂₂ + c₃b₃₂
• x₃ = c₁b₁₃ + c₂b₂₃ + c₃b₃₃

This system represents a set of algebraic equations that we can solve for the unknowns c₁, c₂, and c₃. The number of equations will always match the dimension of the vector space, ensuring that we have enough information to find a unique solution.

Step 3: Solve the System of Linear Equations

There are several methods to solve the system of linear equations obtained in Step 2. Common methods include Gaussian elimination, Gauss-Jordan elimination, and using matrix inverses. The choice of method often depends on the specific problem and the tools available.

• Gaussian Elimination: This method involves transforming the system of equations into an upper triangular form using elementary row operations. The solutions can then be found by back-substitution.
• Gauss-Jordan Elimination: This is an extension of Gaussian elimination that transforms the system into reduced row-echelon form, making the solutions immediately apparent.
• Matrix Inverses: If the system can be written in matrix form as Ax = b, where A is the coefficient matrix, x is the vector of unknowns, and b is the constant vector, then the solution can be found by x = A^-1b, provided that A is invertible.

The method of choice will depend on the structure of the system and the available computational resources. Gaussian elimination and Gauss-Jordan elimination are generally applicable and efficient for smaller systems, while matrix inverses can be more efficient for larger systems if the inverse is already known or can be computed easily.

Step 4: Form the Coordinate Matrix

Once we have found the values of c₁, c₂, ..., c_n, we can form the coordinate matrix [x]_B'. The coordinate matrix is simply a column vector whose entries are the scalars we found in Step 3:

[x]_B' =

$$\begin{bmatrix} c_1 \\ c_2 \\ ... \\ c_n \end{bmatrix}$$

This matrix represents the coordinates of the vector x relative to the basis B'. It is a crucial link between the vector x in the standard basis and its representation in the B' basis. The coordinate matrix allows us to perform transformations and computations in the B' basis, which can often simplify complex problems.

By following these four steps, we can systematically find the coordinate matrix of any vector x relative to a given basis B'. This method is a cornerstone of linear algebra and is essential for understanding how vectors behave in different coordinate systems. In the next section, we will apply this method to the specific example provided, illustrating each step in detail.

Example: Finding the Coordinate Matrix

To solidify our understanding, let's apply the step-by-step method to the given example. We are tasked with finding the coordinate matrix of x = (-11,19,-18) in R³ relative to the basis B' = {(8,11,0), (7,0,10), (1,4,6)}.

Step 1: Express x as a Linear Combination of Basis Vectors

We need to find scalars c₁, c₂, and c₃ such that:

(-11,19,-18) = c₁(8,11,0) + c₂(7,0,10) + c₃(1,4,6)

This equation sets up the foundation for our solution. It represents the vector x as a linear combination of the basis vectors in B'. Our goal is to find the unique values of c₁, c₂, and c₃ that satisfy this equation. The existence of such a solution is guaranteed because B' is a basis for R³.

Step 2: Set Up the System of Linear Equations

Breaking the vector equation into its components, we get the following system of linear equations:

• -11 = 8c₁ + 7c₂ + c₃
• 19 = 11c₁ + 0c₂ + 4c₃
• -18 = 0c₁ + 10c₂ + 6c₃

This system of equations represents the algebraic relationships between the components of x and the basis vectors in B'. Each equation corresponds to one dimension in R³, and the coefficients of the c_i variables are derived from the components of the basis vectors. Solving this system will give us the coordinates of x in the B' basis.

Step 3: Solve the System of Linear Equations

We can solve this system using various methods. Here, we will use Gaussian elimination. First, let's write the augmented matrix:

$$\begin{bmatrix} 8 & 7 & 1 & -11 \\ 11 & 0 & 4 & 19 \\ 0 & 10 & 6 & -18 \end{bmatrix}$$

To make the calculations easier, we can use a calculator or software that performs matrix operations. After performing Gaussian elimination (or Gauss-Jordan elimination), we find the solutions:

• c₁ = 2
• c₂ = -1
• c₃ = -7

These values are the unique scalars that satisfy our system of equations. They represent the weights of the basis vectors in the linear combination that equals x. The process of Gaussian elimination involves a series of row operations, such as swapping rows, multiplying a row by a scalar, and adding multiples of one row to another, to transform the matrix into an upper triangular or reduced row-echelon form. These operations systematically eliminate variables, allowing us to solve for the unknowns.

Step 4: Form the Coordinate Matrix

Now that we have c₁ = 2, c₂ = -1, and c₃ = -7, we can form the coordinate matrix of x relative to B':

[x]_B' =

$$\begin{bmatrix} 2 \\ -1 \\ -7 \end{bmatrix}$$

This coordinate matrix represents the vector x in the B' basis. It tells us that to express x as a linear combination of the basis vectors in B', we need to multiply the first basis vector by 2, the second by -1, and the third by -7. This representation is unique and provides a new perspective on the vector x.

Thus, the coordinate matrix of x = (-11,19,-18) relative to the basis B' = {(8,11,0), (7,0,10), (1,4,6)} is:

[x]_B' =

$$\begin{bmatrix} 2 \\ -1 \\ -7 \end{bmatrix}$$

This example illustrates the entire process of finding the coordinate matrix, from expressing x as a linear combination to solving the system of equations and forming the final matrix. Understanding this process is essential for working with different bases and transformations in linear algebra.

Conclusion

In this article, we have explored the concept of coordinate matrices and provided a detailed method for finding the coordinate matrix of a vector x in Rⁿ relative to a basis B'. We began with the theoretical background, emphasizing the importance of expressing a vector as a linear combination of basis vectors. We then outlined a step-by-step approach, which includes expressing x as a linear combination, setting up a system of linear equations, solving the system, and forming the coordinate matrix.

Understanding coordinate matrices is crucial in linear algebra because it allows us to represent vectors in different bases. This capability is not just a theoretical exercise; it has significant practical applications in various fields, including computer graphics, data analysis, and engineering. For example, in computer graphics, coordinate matrices are used to transform objects between different coordinate systems, which is essential for rendering 3D scenes. In data analysis, changing bases can simplify data representation and highlight important features.

We illustrated the method with a specific example, demonstrating each step in detail. By working through the example, we showed how to transform the theoretical steps into a concrete calculation. The key takeaway is that finding the coordinate matrix involves solving a system of linear equations, and the solution provides the coordinates of x in the B' basis. This process allows us to switch between different representations of vectors, providing flexibility and efficiency in various applications.

The ability to find coordinate matrices is a valuable skill for anyone working with linear algebra. It provides a deeper understanding of vector spaces and linear transformations. Whether you are a student learning linear algebra or a professional applying these concepts in your field, mastering this skill will undoubtedly enhance your problem-solving capabilities. We encourage readers to practice this method with different examples to further solidify their understanding. With a solid grasp of coordinate matrices, you will be well-equipped to tackle more advanced topics in linear algebra and its applications.

In summary, the coordinate matrix is a fundamental tool in linear algebra, enabling us to represent vectors in different bases and facilitating various transformations and computations. The step-by-step method outlined in this article provides a clear and systematic approach to finding these matrices, making it an essential skill for anyone working with vector spaces and linear transformations.