Find Coordinate Vector With Respect To Basis

Finding Coordinate Vectors with Respect to a Basis

Finding the coordinate vector of a vector with respect to a given basis is a fundamental concept in linear algebra. This article will guide you through the process, explaining the underlying theory and providing practical examples. Understanding this concept is crucial for working with vector spaces and solving problems in various fields, including computer graphics, machine learning, and physics.

What is a Coordinate Vector?

A coordinate vector represents a vector in terms of a specific basis. A basis is a set of linearly independent vectors that span the entire vector space. This means that any vector in the space can be expressed as a unique linear combination of the basis vectors. The coordinate vector simply lists the coefficients of this linear combination.

The Process: A Step-by-Step Guide

Let's assume we have a vector v and a basis B = {b₁, b₂, ..., bₙ} for a vector space V. To find the coordinate vector of v with respect to B, denoted as [v]_B, we need to solve the following equation:

v = c₁b₁ + c₂b₂ + ... + cₙbₙ

where c₁, c₂, ..., cₙ are scalar coefficients. The coordinate vector [v]_B is then given by:

[v]_B = [c₁, c₂, ..., cₙ]^T

This equation represents a system of linear equations. To solve it, we can use various methods, including Gaussian elimination or matrix inversion. Let's illustrate with examples.

Example 1: A Simple 2D Case

Let's say we have the vector v = [3, 5]^T and the basis B = {[1, 0]^T, [0, 1]^T} (the standard basis). The equation becomes:

[3, 5]^T = c₁[1, 0]^T + c₂[0, 1]^T

This simplifies to:

3 = c₁ 5 = c₂

Therefore, the coordinate vector of v with respect to B is [3, 5]^T. This is unsurprising, as we're using the standard basis.

Example 2: A Non-Standard Basis

Now, let's consider a different basis. Let v = [3, 5]^T and B = {[1, 1]^T, [1, -1]^T}. Our equation is:

[3, 5]^T = c₁[1, 1]^T + c₂[1, -1]^T

This leads to the system of equations:

c₁ + c₂ = 3 c₁ - c₂ = 5

Solving this system (e.g., by adding the two equations), we get c₁ = 4 and c₂ = -1. Therefore, the coordinate vector of v with respect to B is [4, -1]^T.

Example 3: Higher Dimensions

The same principle applies to higher dimensions. Consider a 3D vector and a basis with three vectors. The system of equations will simply have three equations and three unknowns, which can be solved using similar methods. For instance, matrix methods like row reduction are particularly efficient for solving such systems in higher dimensions.

Using Matrices for Efficient Calculation

For larger systems, it's more efficient to use matrices. We can represent the basis vectors as columns of a matrix A, and the coordinate vector as a column vector x. Then, the equation becomes:

Ax = v

Solving for x, we obtain the coordinate vector:

x = A⁻¹v (provided A is invertible)

Conclusion

Finding the coordinate vector of a vector with respect to a given basis is a fundamental linear algebra operation with wide-ranging applications. Understanding the process, whether through solving systems of equations directly or using matrix methods, is key to mastering linear algebra and its applications in various fields. Remember to always ensure your basis vectors are linearly independent to form a valid basis. This ensures a unique solution for the coordinate vector.