Change of Basis in 2D Linear Algebra: Translating Vectors Between Bases

Overview

In this video, the speaker explains that a vector’s coordinates depend on the chosen basis and demonstrates how to translate a vector between two different bases using a change of basis matrix. A concrete 2D example contrasts the standard basis with Jennifer’s basis, showing how a vector described as (3,2) in one basis becomes (5/3,1/3) in another, and how to reverse the process with the inverse matrix. The discussion also links coordinate changes to linear transformations and how to compose them across bases.

• Coordinates depend on the basis, not on the vector alone
• The change of basis matrix translates coordinates from one basis to another
• The inverse change of basis matrix returns to the original language
• A simple 2D rotation can be represented differently in different bases

Introduction

This video introduces the core idea that coordinates are defined relative to a chosen basis. It uses a 2D setting to show how the same vector can have different coordinate pairs when described in different bases, underscoring the dependence of coordinates on basis choice.

Coordinate Systems and Basis Vectors

The standard 2D coordinate system uses unit basis vectors I hat and J hat, pointing right and up. A vector with coordinates (3, 2) is obtained by scaling these basis directions and adding the results. Jennifer operates in a different language, with basis vectors B1 and B2 that point in other directions. In Jennifer’s world, the same vector is described by another pair of coordinates, illustrating how two different bases encode the same geometry with different numerical descriptions.

Change of Basis and Matrix Representation

The central tool for translating between bases is the change of basis matrix. If we write Jennifer’s basis vectors as columns of a matrix, multiplying this matrix by a vector described in Jennifer’s language yields the corresponding vector in our language. The inverse matrix performs the reverse translation. The columns of the change of basis matrix encode how the new basis sits inside the old coordinate frame, tying the two descriptions together.

A Concrete 2D Example

Consider Jennifer’s basis vectors B1 = (2, 1) and B2 = (-1, 1) when written in our coordinates. To express our vector v = (3, 2) in Jennifer’s coordinates, we solve a and b in a·B1 + b·B2 = v. This yields a = 5/3 and b = 1/3, so Jennifer would describe v as (5/3, 1/3). Conversely, using the inverse of the basis matrix, Jennifer’s coordinates (5/3, 1/3) map back to (3, 2) in our language. This example makes explicit how the change of basis matrix works and why the inverse is needed for the opposite translation.

Transformations Across Bases

Extending beyond individual vectors, a change of basis can be viewed as a linear transformation that moves our grid into Jennifer’s grid. When we describe rotations or other transformations, we must express them in Jennifer’s language to describe how they act on she describes basis vectors. The standard procedure is to translate a vector from Jennifer’s language to ours, apply the transformation in our coordinates, and then translate back to Jennifer’s language. The resulting matrix, formed by composing these three steps, represents the transformation in Jennifer’s basis. In the worked example, applying this three-matrix product yields a rotation described in Jennifer’s coordinates.

Conclusion

The talk closes by highlighting how these ideas underpin more advanced topics in linear algebra, such as eigenvectors and eigenvalues, which will be explored in a future video. The central message is that alternate coordinate systems illuminate different perspectives on the same linear transformations.