Find Coordinate Vector With Respect To Basis

Finding Coordinate Vectors with Respect to a Basis: A Comprehensive Guide

Finding the coordinate vector of a vector with respect to a given basis is a fundamental concept in linear algebra. Understanding this process is crucial for mastering more advanced topics like linear transformations, eigenvalues, and eigenvectors. This article will provide a comprehensive guide, walking you through the process step-by-step, explaining the underlying theory, and addressing common questions. We'll explore various methods and provide examples to solidify your understanding. The key concept is understanding how to express a vector as a linear combination of the basis vectors.

Introduction: What are Coordinate Vectors and Bases?

Before diving into the mechanics of finding coordinate vectors, let's establish a firm understanding of the underlying concepts. A vector is a mathematical object that has both magnitude and direction. In linear algebra, we often represent vectors as ordered lists of numbers (e.g., [1, 2, 3]). A basis for a vector space is a set of linearly independent vectors that span the entire space. This means that any vector in the space can be expressed as a unique linear combination of the basis vectors.

The coordinate vector of a vector with respect to a particular basis is simply the set of scalars that, when multiplied by the basis vectors and summed, yield the original vector. These scalars represent the "coordinates" of the vector in the coordinate system defined by the basis. Think of it like this: the basis vectors define the axes of your coordinate system, and the coordinate vector tells you how far along each axis you need to go to reach the specific vector.

Method 1: Solving a System of Linear Equations

This is the most straightforward method, especially for beginners. Let's say we have a vector v and a basis B = {b₁, b₂, ..., bₙ} for an n-dimensional vector space. We want to find the coordinate vector [v]_B, which represents v with respect to basis B. This means we want to find scalars c₁, c₂, ..., cₙ such that:

v = c₁b₁ + c₂b₂ + ... + cₙbₙ

This equation represents a system of linear equations. If the vectors are represented as column vectors, the equation can be written in matrix form as:

[b₁ b₂ ... bₙ] [c₁] = [v] [c₂] [...] [cₙ]

The matrix [b₁ b₂ ... bₙ] is formed by placing the basis vectors as columns. This matrix is often called the change-of-basis matrix or, if it's a square matrix, the transition matrix. Solving this system of equations (e.g., using Gaussian elimination or matrix inversion) will yield the values of c₁, c₂, ..., cₙ, which form the coordinate vector [v]_B = [c₁, c₂, ..., cₙ]ᵀ.

Example:

Let's say we have the vector v = [5, 7] and the basis B = {[1, 2], [3, 1]}. We want to find [v]_B. The system of equations is:

c₁[1] + c₂[3] = [5] c₁[2] + c₂[1] = [7]

This can be written in matrix form as:

[[1, 3], [2, 1]] [[c₁], [c₂]] = [[5], [7]]

Solving this system (e.g., using Gaussian elimination) gives c₁ = 1 and c₂ = 4/3. Therefore, the coordinate vector is [v]_B = [1, 4/3]. This means that v = 1*[1,2] + (4/3)*[3,1].

Method 2: Using the Inverse of the Change-of-Basis Matrix

If the basis vectors form a square matrix (meaning we're dealing with a square matrix and a full basis), we can use matrix inversion to find the coordinate vector more directly. Let's denote the matrix formed by the basis vectors as P. Then the equation becomes:

P[v]_B = v

To find the coordinate vector, we simply multiply both sides by the inverse of P, provided the inverse exists (which means the basis vectors are linearly independent):

[v]_B = P⁻¹v

Example:

Using the same example as before, P = [[1, 3], [2, 1]]. The inverse of P is calculated as P⁻¹ = [-1/5, 3/5; 2/5, -1/5]. Therefore:

[v]_B = P⁻¹v = [-1/5, 3/5; 2/5, -1/5] * [5; 7] = [1; 4/3]

This method is computationally efficient for larger matrices, provided the inverse can be readily computed. Numerical computation packages and tools easily handle these matrix operations.

Method 3: Gram-Schmidt Process (For Non-Orthogonal Bases)

When dealing with non-orthogonal bases, the above methods still apply. However, working with non-orthogonal vectors can lead to more complex calculations. The Gram-Schmidt process offers a way to orthogonalize a basis, which simplifies subsequent calculations. This process systematically transforms a set of linearly independent vectors into an orthonormal set (orthogonal vectors with unit length). While this doesn't directly give you the coordinate vector, it simplifies the calculation if you then use the orthogonal basis with Method 1 or 2.

The Gram-Schmidt process involves projecting vectors onto the orthogonal complement of the subspace spanned by the previously orthogonalized vectors. This recursive process guarantees an orthogonal set.

Example: While a detailed explanation of the Gram-Schmidt process is beyond the scope of this brief section, the general approach involves iteratively making vectors orthogonal. After creating an orthonormal basis, you can reapply Method 1 or 2 using the new, easier to work with, orthogonal basis.

The Importance of Linear Independence

It's crucial to remember that the basis vectors must be linearly independent. If they are linearly dependent (meaning one vector can be expressed as a linear combination of the others), they do not form a valid basis, and the methods described above will fail. This is because a unique solution for the coordinate vector cannot be guaranteed.

Explanation of the underlying linear algebra

The core concept is that a vector space is a set of vectors that can be added together and multiplied by scalars, and these operations are closed in the set. A basis provides a coordinate system for this vector space. Each basis vector represents an axis in the space. To find a coordinate vector, you're essentially asking: "How many units along each axis do we need to travel to reach the vector in question?"

The system of linear equations we solve arises directly from the definition of a linear combination. The matrix form is simply a compact representation of that system. The success of these methods relies on the fact that a basis provides a unique representation of any vector in the space.

Frequently Asked Questions (FAQ)

Q: What happens if the basis vectors are not linearly independent?

A: If the basis vectors are linearly dependent, they do not span the entire vector space, and you cannot uniquely express every vector in the space as a linear combination of those vectors. This will result in either no solution or infinitely many solutions to the system of equations. In essence, it's not a valid basis.

Q: Can I use any set of vectors as a basis?

A: No, only sets of linearly independent vectors that span the entire vector space can serve as a basis. The number of vectors in the basis must equal the dimension of the vector space.

Q: What if my vector space is infinite-dimensional?

A: The concept of coordinate vectors still applies, but the basis will be infinite, and the methods for finding coordinate vectors become more abstract and involve concepts from functional analysis.

Q: What is the significance of the coordinate vector?

A: Coordinate vectors are fundamental for many operations in linear algebra. They allow you to represent vectors in different coordinate systems, simplifying calculations involving linear transformations. They are also essential for understanding concepts like eigenvalues and eigenvectors.

Q: How do I choose a basis?

A: The choice of basis often depends on the specific problem or application. Sometimes a standard basis (e.g., the canonical basis consisting of vectors with a single 1 and the rest zeros) is used for convenience. Other times, a basis that is tailored to the problem's structure (e.g., an orthogonal basis) might be more efficient.

Conclusion

Finding the coordinate vector of a vector with respect to a given basis is a core skill in linear algebra. Understanding this concept opens the door to a deeper understanding of linear transformations, matrix operations, and other advanced topics. While the process involves solving systems of linear equations, the underlying concept is simply expressing a vector as a unique linear combination of basis vectors. By mastering this fundamental concept, you'll be well-equipped to tackle more complex problems in linear algebra and its various applications. Remember to always check for linear independence of your basis vectors to ensure the validity of your results. The choice of method depends on the nature of the basis (orthogonal or non-orthogonal) and the computational tools at your disposal. With practice and a solid grasp of the underlying theory, finding coordinate vectors will become second nature.